Metamath Proof Explorer


Theorem naddcllem

Description: Lemma for ordinal addition closure. (Contributed by Scott Fenton, 26-Aug-2024)

Ref Expression
Assertion naddcllem ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +no 𝐵 ) ∈ On ∧ ( 𝐴 +no 𝐵 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) )

Proof

Step Hyp Ref Expression
1 oveq1 ( 𝑎 = 𝑐 → ( 𝑎 +no 𝑏 ) = ( 𝑐 +no 𝑏 ) )
2 1 eleq1d ( 𝑎 = 𝑐 → ( ( 𝑎 +no 𝑏 ) ∈ On ↔ ( 𝑐 +no 𝑏 ) ∈ On ) )
3 sneq ( 𝑎 = 𝑐 → { 𝑎 } = { 𝑐 } )
4 3 xpeq1d ( 𝑎 = 𝑐 → ( { 𝑎 } × 𝑏 ) = ( { 𝑐 } × 𝑏 ) )
5 4 imaeq2d ( 𝑎 = 𝑐 → ( +no “ ( { 𝑎 } × 𝑏 ) ) = ( +no “ ( { 𝑐 } × 𝑏 ) ) )
6 5 sseq1d ( 𝑎 = 𝑐 → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ) )
7 xpeq1 ( 𝑎 = 𝑐 → ( 𝑎 × { 𝑏 } ) = ( 𝑐 × { 𝑏 } ) )
8 7 imaeq2d ( 𝑎 = 𝑐 → ( +no “ ( 𝑎 × { 𝑏 } ) ) = ( +no “ ( 𝑐 × { 𝑏 } ) ) )
9 8 sseq1d ( 𝑎 = 𝑐 → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) )
10 6 9 anbi12d ( 𝑎 = 𝑐 → ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) ) )
11 10 rabbidv ( 𝑎 = 𝑐 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
12 11 inteqd ( 𝑎 = 𝑐 { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
13 1 12 eqeq12d ( 𝑎 = 𝑐 → ( ( 𝑎 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ↔ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) )
14 2 13 anbi12d ( 𝑎 = 𝑐 → ( ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) )
15 oveq2 ( 𝑏 = 𝑑 → ( 𝑐 +no 𝑏 ) = ( 𝑐 +no 𝑑 ) )
16 15 eleq1d ( 𝑏 = 𝑑 → ( ( 𝑐 +no 𝑏 ) ∈ On ↔ ( 𝑐 +no 𝑑 ) ∈ On ) )
17 xpeq2 ( 𝑏 = 𝑑 → ( { 𝑐 } × 𝑏 ) = ( { 𝑐 } × 𝑑 ) )
18 17 imaeq2d ( 𝑏 = 𝑑 → ( +no “ ( { 𝑐 } × 𝑏 ) ) = ( +no “ ( { 𝑐 } × 𝑑 ) ) )
19 18 sseq1d ( 𝑏 = 𝑑 → ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ) )
20 sneq ( 𝑏 = 𝑑 → { 𝑏 } = { 𝑑 } )
21 20 xpeq2d ( 𝑏 = 𝑑 → ( 𝑐 × { 𝑏 } ) = ( 𝑐 × { 𝑑 } ) )
22 21 imaeq2d ( 𝑏 = 𝑑 → ( +no “ ( 𝑐 × { 𝑏 } ) ) = ( +no “ ( 𝑐 × { 𝑑 } ) ) )
23 22 sseq1d ( 𝑏 = 𝑑 → ( ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) )
24 19 23 anbi12d ( 𝑏 = 𝑑 → ( ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) ) )
25 24 rabbidv ( 𝑏 = 𝑑 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } )
26 25 inteqd ( 𝑏 = 𝑑 { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } )
27 15 26 eqeq12d ( 𝑏 = 𝑑 → ( ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ↔ ( 𝑐 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) )
28 16 27 anbi12d ( 𝑏 = 𝑑 → ( ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) )
29 oveq1 ( 𝑎 = 𝑐 → ( 𝑎 +no 𝑑 ) = ( 𝑐 +no 𝑑 ) )
30 29 eleq1d ( 𝑎 = 𝑐 → ( ( 𝑎 +no 𝑑 ) ∈ On ↔ ( 𝑐 +no 𝑑 ) ∈ On ) )
31 3 xpeq1d ( 𝑎 = 𝑐 → ( { 𝑎 } × 𝑑 ) = ( { 𝑐 } × 𝑑 ) )
32 31 imaeq2d ( 𝑎 = 𝑐 → ( +no “ ( { 𝑎 } × 𝑑 ) ) = ( +no “ ( { 𝑐 } × 𝑑 ) ) )
33 32 sseq1d ( 𝑎 = 𝑐 → ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ) )
34 xpeq1 ( 𝑎 = 𝑐 → ( 𝑎 × { 𝑑 } ) = ( 𝑐 × { 𝑑 } ) )
35 34 imaeq2d ( 𝑎 = 𝑐 → ( +no “ ( 𝑎 × { 𝑑 } ) ) = ( +no “ ( 𝑐 × { 𝑑 } ) ) )
36 35 sseq1d ( 𝑎 = 𝑐 → ( ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) )
37 33 36 anbi12d ( 𝑎 = 𝑐 → ( ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) ) )
38 37 rabbidv ( 𝑎 = 𝑐 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } )
39 38 inteqd ( 𝑎 = 𝑐 { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } )
40 29 39 eqeq12d ( 𝑎 = 𝑐 → ( ( 𝑎 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ↔ ( 𝑐 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) )
41 30 40 anbi12d ( 𝑎 = 𝑐 → ( ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) )
42 oveq1 ( 𝑎 = 𝐴 → ( 𝑎 +no 𝑏 ) = ( 𝐴 +no 𝑏 ) )
43 42 eleq1d ( 𝑎 = 𝐴 → ( ( 𝑎 +no 𝑏 ) ∈ On ↔ ( 𝐴 +no 𝑏 ) ∈ On ) )
44 sneq ( 𝑎 = 𝐴 → { 𝑎 } = { 𝐴 } )
45 44 xpeq1d ( 𝑎 = 𝐴 → ( { 𝑎 } × 𝑏 ) = ( { 𝐴 } × 𝑏 ) )
46 45 imaeq2d ( 𝑎 = 𝐴 → ( +no “ ( { 𝑎 } × 𝑏 ) ) = ( +no “ ( { 𝐴 } × 𝑏 ) ) )
47 46 sseq1d ( 𝑎 = 𝐴 → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ) )
48 xpeq1 ( 𝑎 = 𝐴 → ( 𝑎 × { 𝑏 } ) = ( 𝐴 × { 𝑏 } ) )
49 48 imaeq2d ( 𝑎 = 𝐴 → ( +no “ ( 𝑎 × { 𝑏 } ) ) = ( +no “ ( 𝐴 × { 𝑏 } ) ) )
50 49 sseq1d ( 𝑎 = 𝐴 → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) )
51 47 50 anbi12d ( 𝑎 = 𝐴 → ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) ) )
52 51 rabbidv ( 𝑎 = 𝐴 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
53 52 inteqd ( 𝑎 = 𝐴 { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
54 42 53 eqeq12d ( 𝑎 = 𝐴 → ( ( 𝑎 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ↔ ( 𝐴 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) )
55 43 54 anbi12d ( 𝑎 = 𝐴 → ( ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝐴 +no 𝑏 ) ∈ On ∧ ( 𝐴 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) )
56 oveq2 ( 𝑏 = 𝐵 → ( 𝐴 +no 𝑏 ) = ( 𝐴 +no 𝐵 ) )
57 56 eleq1d ( 𝑏 = 𝐵 → ( ( 𝐴 +no 𝑏 ) ∈ On ↔ ( 𝐴 +no 𝐵 ) ∈ On ) )
58 xpeq2 ( 𝑏 = 𝐵 → ( { 𝐴 } × 𝑏 ) = ( { 𝐴 } × 𝐵 ) )
59 58 imaeq2d ( 𝑏 = 𝐵 → ( +no “ ( { 𝐴 } × 𝑏 ) ) = ( +no “ ( { 𝐴 } × 𝐵 ) ) )
60 59 sseq1d ( 𝑏 = 𝐵 → ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ) )
61 sneq ( 𝑏 = 𝐵 → { 𝑏 } = { 𝐵 } )
62 61 xpeq2d ( 𝑏 = 𝐵 → ( 𝐴 × { 𝑏 } ) = ( 𝐴 × { 𝐵 } ) )
63 62 imaeq2d ( 𝑏 = 𝐵 → ( +no “ ( 𝐴 × { 𝑏 } ) ) = ( +no “ ( 𝐴 × { 𝐵 } ) ) )
64 63 sseq1d ( 𝑏 = 𝐵 → ( ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) )
65 60 64 anbi12d ( 𝑏 = 𝐵 → ( ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) ) )
66 65 rabbidv ( 𝑏 = 𝐵 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } )
67 66 inteqd ( 𝑏 = 𝐵 { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } )
68 56 67 eqeq12d ( 𝑏 = 𝐵 → ( ( 𝐴 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ↔ ( 𝐴 +no 𝐵 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) )
69 57 68 anbi12d ( 𝑏 = 𝐵 → ( ( ( 𝐴 +no 𝑏 ) ∈ On ∧ ( 𝐴 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝐴 +no 𝐵 ) ∈ On ∧ ( 𝐴 +no 𝐵 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) ) )
70 simpl ( ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) → ( 𝑐 +no 𝑏 ) ∈ On )
71 70 ralimi ( ∀ 𝑐𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) → ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On )
72 71 3ad2ant2 ( ( ∀ 𝑐𝑎𝑑𝑏 ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑐𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑑𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) → ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On )
73 simpl ( ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) → ( 𝑎 +no 𝑑 ) ∈ On )
74 73 ralimi ( ∀ 𝑑𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) → ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On )
75 74 3ad2ant3 ( ( ∀ 𝑐𝑎𝑑𝑏 ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑐𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑑𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) → ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On )
76 72 75 jca ( ( ∀ 𝑐𝑎𝑑𝑏 ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑐𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑑𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) → ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) )
77 df-nadd +no = frecs ( { ⟨ 𝑝 , 𝑞 ⟩ ∣ ( 𝑝 ∈ ( On × On ) ∧ 𝑞 ∈ ( On × On ) ∧ ( ( ( 1st𝑝 ) E ( 1st𝑞 ) ∨ ( 1st𝑝 ) = ( 1st𝑞 ) ) ∧ ( ( 2nd𝑝 ) E ( 2nd𝑞 ) ∨ ( 2nd𝑝 ) = ( 2nd𝑞 ) ) ∧ 𝑝𝑞 ) ) } , ( On × On ) , ( 𝑡 ∈ V , 𝑓 ∈ V ↦ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } ) )
78 77 on2recsov ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( 𝑎 +no 𝑏 ) = ( ⟨ 𝑎 , 𝑏 ⟩ ( 𝑡 ∈ V , 𝑓 ∈ V ↦ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } ) ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) ) )
79 78 adantr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 +no 𝑏 ) = ( ⟨ 𝑎 , 𝑏 ⟩ ( 𝑡 ∈ V , 𝑓 ∈ V ↦ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } ) ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) ) )
80 opex 𝑎 , 𝑏 ⟩ ∈ V
81 naddfn +no Fn ( On × On )
82 fnfun ( +no Fn ( On × On ) → Fun +no )
83 81 82 ax-mp Fun +no
84 vex 𝑎 ∈ V
85 84 sucex suc 𝑎 ∈ V
86 vex 𝑏 ∈ V
87 86 sucex suc 𝑏 ∈ V
88 85 87 xpex ( suc 𝑎 × suc 𝑏 ) ∈ V
89 88 difexi ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ∈ V
90 resfunexg ( ( Fun +no ∧ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ∈ V ) → ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) ∈ V )
91 83 89 90 mp2an ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) ∈ V
92 eloni ( 𝑏 ∈ On → Ord 𝑏 )
93 92 ad2antlr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → Ord 𝑏 )
94 ordirr ( Ord 𝑏 → ¬ 𝑏𝑏 )
95 93 94 syl ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ¬ 𝑏𝑏 )
96 95 olcd ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ¬ 𝑎 ∈ { 𝑎 } ∨ ¬ 𝑏𝑏 ) )
97 ianor ( ¬ ( 𝑎 ∈ { 𝑎 } ∧ 𝑏𝑏 ) ↔ ( ¬ 𝑎 ∈ { 𝑎 } ∨ ¬ 𝑏𝑏 ) )
98 opelxp ( ⟨ 𝑎 , 𝑏 ⟩ ∈ ( { 𝑎 } × 𝑏 ) ↔ ( 𝑎 ∈ { 𝑎 } ∧ 𝑏𝑏 ) )
99 97 98 xchnxbir ( ¬ ⟨ 𝑎 , 𝑏 ⟩ ∈ ( { 𝑎 } × 𝑏 ) ↔ ( ¬ 𝑎 ∈ { 𝑎 } ∨ ¬ 𝑏𝑏 ) )
100 96 99 sylibr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ¬ ⟨ 𝑎 , 𝑏 ⟩ ∈ ( { 𝑎 } × 𝑏 ) )
101 84 sucid 𝑎 ∈ suc 𝑎
102 snssi ( 𝑎 ∈ suc 𝑎 → { 𝑎 } ⊆ suc 𝑎 )
103 101 102 ax-mp { 𝑎 } ⊆ suc 𝑎
104 sssucid 𝑏 ⊆ suc 𝑏
105 xpss12 ( ( { 𝑎 } ⊆ suc 𝑎𝑏 ⊆ suc 𝑏 ) → ( { 𝑎 } × 𝑏 ) ⊆ ( suc 𝑎 × suc 𝑏 ) )
106 103 104 105 mp2an ( { 𝑎 } × 𝑏 ) ⊆ ( suc 𝑎 × suc 𝑏 )
107 ssdifsn ( ( { 𝑎 } × 𝑏 ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ↔ ( ( { 𝑎 } × 𝑏 ) ⊆ ( suc 𝑎 × suc 𝑏 ) ∧ ¬ ⟨ 𝑎 , 𝑏 ⟩ ∈ ( { 𝑎 } × 𝑏 ) ) )
108 106 107 mpbiran ( ( { 𝑎 } × 𝑏 ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ↔ ¬ ⟨ 𝑎 , 𝑏 ⟩ ∈ ( { 𝑎 } × 𝑏 ) )
109 100 108 sylibr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( { 𝑎 } × 𝑏 ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) )
110 resima2 ( ( { 𝑎 } × 𝑏 ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) → ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) = ( +no “ ( { 𝑎 } × 𝑏 ) ) )
111 109 110 syl ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) = ( +no “ ( { 𝑎 } × 𝑏 ) ) )
112 111 sseq1d ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ) )
113 eloni ( 𝑎 ∈ On → Ord 𝑎 )
114 113 ad2antrr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → Ord 𝑎 )
115 ordirr ( Ord 𝑎 → ¬ 𝑎𝑎 )
116 114 115 syl ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ¬ 𝑎𝑎 )
117 116 orcd ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ¬ 𝑎𝑎 ∨ ¬ 𝑏 ∈ { 𝑏 } ) )
118 ianor ( ¬ ( 𝑎𝑎𝑏 ∈ { 𝑏 } ) ↔ ( ¬ 𝑎𝑎 ∨ ¬ 𝑏 ∈ { 𝑏 } ) )
119 opelxp ( ⟨ 𝑎 , 𝑏 ⟩ ∈ ( 𝑎 × { 𝑏 } ) ↔ ( 𝑎𝑎𝑏 ∈ { 𝑏 } ) )
120 118 119 xchnxbir ( ¬ ⟨ 𝑎 , 𝑏 ⟩ ∈ ( 𝑎 × { 𝑏 } ) ↔ ( ¬ 𝑎𝑎 ∨ ¬ 𝑏 ∈ { 𝑏 } ) )
121 117 120 sylibr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ¬ ⟨ 𝑎 , 𝑏 ⟩ ∈ ( 𝑎 × { 𝑏 } ) )
122 sssucid 𝑎 ⊆ suc 𝑎
123 86 sucid 𝑏 ∈ suc 𝑏
124 snssi ( 𝑏 ∈ suc 𝑏 → { 𝑏 } ⊆ suc 𝑏 )
125 123 124 ax-mp { 𝑏 } ⊆ suc 𝑏
126 xpss12 ( ( 𝑎 ⊆ suc 𝑎 ∧ { 𝑏 } ⊆ suc 𝑏 ) → ( 𝑎 × { 𝑏 } ) ⊆ ( suc 𝑎 × suc 𝑏 ) )
127 122 125 126 mp2an ( 𝑎 × { 𝑏 } ) ⊆ ( suc 𝑎 × suc 𝑏 )
128 ssdifsn ( ( 𝑎 × { 𝑏 } ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ↔ ( ( 𝑎 × { 𝑏 } ) ⊆ ( suc 𝑎 × suc 𝑏 ) ∧ ¬ ⟨ 𝑎 , 𝑏 ⟩ ∈ ( 𝑎 × { 𝑏 } ) ) )
129 127 128 mpbiran ( ( 𝑎 × { 𝑏 } ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ↔ ¬ ⟨ 𝑎 , 𝑏 ⟩ ∈ ( 𝑎 × { 𝑏 } ) )
130 121 129 sylibr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 × { 𝑏 } ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) )
131 resima2 ( ( 𝑎 × { 𝑏 } ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) → ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) = ( +no “ ( 𝑎 × { 𝑏 } ) ) )
132 130 131 syl ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) = ( +no “ ( 𝑎 × { 𝑏 } ) ) )
133 132 sseq1d ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) )
134 112 133 anbi12d ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) )
135 134 rabbidv ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
136 135 inteqd ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
137 simprr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On )
138 oveq1 ( 𝑡 = 𝑎 → ( 𝑡 +no 𝑑 ) = ( 𝑎 +no 𝑑 ) )
139 138 eleq1d ( 𝑡 = 𝑎 → ( ( 𝑡 +no 𝑑 ) ∈ On ↔ ( 𝑎 +no 𝑑 ) ∈ On ) )
140 139 ralbidv ( 𝑡 = 𝑎 → ( ∀ 𝑑𝑏 ( 𝑡 +no 𝑑 ) ∈ On ↔ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) )
141 84 140 ralsn ( ∀ 𝑡 ∈ { 𝑎 } ∀ 𝑑𝑏 ( 𝑡 +no 𝑑 ) ∈ On ↔ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On )
142 137 141 sylibr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∀ 𝑡 ∈ { 𝑎 } ∀ 𝑑𝑏 ( 𝑡 +no 𝑑 ) ∈ On )
143 snssi ( 𝑎 ∈ On → { 𝑎 } ⊆ On )
144 onss ( 𝑏 ∈ On → 𝑏 ⊆ On )
145 xpss12 ( ( { 𝑎 } ⊆ On ∧ 𝑏 ⊆ On ) → ( { 𝑎 } × 𝑏 ) ⊆ ( On × On ) )
146 143 144 145 syl2an ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( { 𝑎 } × 𝑏 ) ⊆ ( On × On ) )
147 146 adantr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( { 𝑎 } × 𝑏 ) ⊆ ( On × On ) )
148 81 fndmi dom +no = ( On × On )
149 147 148 sseqtrrdi ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( { 𝑎 } × 𝑏 ) ⊆ dom +no )
150 funimassov ( ( Fun +no ∧ ( { 𝑎 } × 𝑏 ) ⊆ dom +no ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ On ↔ ∀ 𝑡 ∈ { 𝑎 } ∀ 𝑑𝑏 ( 𝑡 +no 𝑑 ) ∈ On ) )
151 83 149 150 sylancr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ On ↔ ∀ 𝑡 ∈ { 𝑎 } ∀ 𝑑𝑏 ( 𝑡 +no 𝑑 ) ∈ On ) )
152 142 151 mpbird ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ On )
153 simprl ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On )
154 oveq2 ( 𝑡 = 𝑏 → ( 𝑐 +no 𝑡 ) = ( 𝑐 +no 𝑏 ) )
155 154 eleq1d ( 𝑡 = 𝑏 → ( ( 𝑐 +no 𝑡 ) ∈ On ↔ ( 𝑐 +no 𝑏 ) ∈ On ) )
156 86 155 ralsn ( ∀ 𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ↔ ( 𝑐 +no 𝑏 ) ∈ On )
157 156 ralbii ( ∀ 𝑐𝑎𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ↔ ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On )
158 153 157 sylibr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∀ 𝑐𝑎𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On )
159 onss ( 𝑎 ∈ On → 𝑎 ⊆ On )
160 snssi ( 𝑏 ∈ On → { 𝑏 } ⊆ On )
161 xpss12 ( ( 𝑎 ⊆ On ∧ { 𝑏 } ⊆ On ) → ( 𝑎 × { 𝑏 } ) ⊆ ( On × On ) )
162 159 160 161 syl2an ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( 𝑎 × { 𝑏 } ) ⊆ ( On × On ) )
163 162 adantr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 × { 𝑏 } ) ⊆ ( On × On ) )
164 163 148 sseqtrrdi ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 × { 𝑏 } ) ⊆ dom +no )
165 funimassov ( ( Fun +no ∧ ( 𝑎 × { 𝑏 } ) ⊆ dom +no ) → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ On ↔ ∀ 𝑐𝑎𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ) )
166 83 164 165 sylancr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ On ↔ ∀ 𝑐𝑎𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ) )
167 158 166 mpbird ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ On )
168 152 167 unssd ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ On )
169 ssorduni ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ On → Ord ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) )
170 168 169 syl ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → Ord ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) )
171 snex { 𝑎 } ∈ V
172 171 86 xpex ( { 𝑎 } × 𝑏 ) ∈ V
173 funimaexg ( ( Fun +no ∧ ( { 𝑎 } × 𝑏 ) ∈ V ) → ( +no “ ( { 𝑎 } × 𝑏 ) ) ∈ V )
174 83 172 173 mp2an ( +no “ ( { 𝑎 } × 𝑏 ) ) ∈ V
175 snex { 𝑏 } ∈ V
176 84 175 xpex ( 𝑎 × { 𝑏 } ) ∈ V
177 funimaexg ( ( Fun +no ∧ ( 𝑎 × { 𝑏 } ) ∈ V ) → ( +no “ ( 𝑎 × { 𝑏 } ) ) ∈ V )
178 83 176 177 mp2an ( +no “ ( 𝑎 × { 𝑏 } ) ) ∈ V
179 174 178 unex ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ V
180 179 uniex ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ V
181 180 elon ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ↔ Ord ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) )
182 170 181 sylibr ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On )
183 sucelon ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ↔ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On )
184 182 183 sylib ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On )
185 onsucuni ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ On → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) )
186 168 185 syl ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) )
187 186 unssad ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) )
188 186 unssbd ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) )
189 sseq2 ( 𝑥 = suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) )
190 sseq2 ( 𝑥 = suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) )
191 189 190 anbi12d ( 𝑥 = suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) → ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) ) )
192 191 rspcev ( ( suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ∧ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ suc ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) ) → ∃ 𝑥 ∈ On ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) )
193 184 187 188 192 syl12anc ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∃ 𝑥 ∈ On ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) )
194 onintrab2 ( ∃ 𝑥 ∈ On ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ∈ On )
195 193 194 sylib ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ∈ On )
196 136 195 eqeltrd ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ∈ On )
197 84 86 op1std ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( 1st𝑡 ) = 𝑎 )
198 197 sneqd ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → { ( 1st𝑡 ) } = { 𝑎 } )
199 84 86 op2ndd ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( 2nd𝑡 ) = 𝑏 )
200 198 199 xpeq12d ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) = ( { 𝑎 } × 𝑏 ) )
201 200 imaeq2d ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) = ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) )
202 201 sseq1d ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ↔ ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ) )
203 199 sneqd ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → { ( 2nd𝑡 ) } = { 𝑏 } )
204 197 203 xpeq12d ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) = ( 𝑎 × { 𝑏 } ) )
205 204 imaeq2d ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) = ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) )
206 205 sseq1d ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ↔ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) )
207 202 206 anbi12d ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → ( ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) ↔ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) )
208 207 rabbidv ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
209 208 inteqd ( 𝑡 = ⟨ 𝑎 , 𝑏 ⟩ → { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
210 imaeq1 ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) → ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) = ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) )
211 210 sseq1d ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) → ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ) )
212 imaeq1 ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) → ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) = ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) )
213 212 sseq1d ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) → ( ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) )
214 211 213 anbi12d ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) → ( ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) )
215 214 rabbidv ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) → { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
216 215 inteqd ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) → { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
217 eqid ( 𝑡 ∈ V , 𝑓 ∈ V ↦ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } ) = ( 𝑡 ∈ V , 𝑓 ∈ V ↦ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } )
218 209 216 217 ovmpog ( ( ⟨ 𝑎 , 𝑏 ⟩ ∈ V ∧ ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) ∈ V ∧ { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ∈ On ) → ( ⟨ 𝑎 , 𝑏 ⟩ ( 𝑡 ∈ V , 𝑓 ∈ V ↦ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } ) ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) ) = { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
219 80 91 196 218 mp3an12i ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ⟨ 𝑎 , 𝑏 ⟩ ( 𝑡 ∈ V , 𝑓 ∈ V ↦ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st𝑡 ) } × ( 2nd𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st𝑡 ) × { ( 2nd𝑡 ) } ) ) ⊆ 𝑥 ) } ) ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) ) = { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { ⟨ 𝑎 , 𝑏 ⟩ } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
220 79 219 136 3eqtrd ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } )
221 220 195 eqeltrd ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 +no 𝑏 ) ∈ On )
222 221 220 jca ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) )
223 222 ex ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( ( ∀ 𝑐𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) → ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) )
224 76 223 syl5 ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( ( ∀ 𝑐𝑎𝑑𝑏 ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑐𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑑𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) → ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) )
225 14 28 41 55 69 224 on2ind ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +no 𝐵 ) ∈ On ∧ ( 𝐴 +no 𝐵 ) = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) )