Metamath Proof Explorer

Theorem ltsval2

Description: Alternate expression for surreal less-than. Two surreals obey surreal less-than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011)

Ref Expression
Assertion ltsval2 ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )

Proof

Step Hyp Ref Expression
1 ltsval ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) ) )
2 fvex ⊢ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ V
3 fvex ⊢ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ V
4 2 3 brtp ⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ↔ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) )
5 1n0 ⊢ 1o ≠ ∅
6 5 neii ⊢ ¬ 1o = ∅
7 eqeq1 ⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ↔ 1o = ∅ ) )
8 6 7 mtbiri ⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o → ¬ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ )
9 fvprc ⊢ ( ¬ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ )
10 8 9 nsyl2 ⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V )
11 10 adantr ⊢ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V )
12 10 adantr ⊢ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V )
13 2on0 ⊢ 2o ≠ ∅
14 13 neii ⊢ ¬ 2o = ∅
15 eqeq1 ⊢ ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o → ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ↔ 2o = ∅ ) )
16 14 15 mtbiri ⊢ ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o → ¬ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ )
17 fvprc ⊢ ( ¬ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ )
18 16 17 nsyl2 ⊢ ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V )
19 18 adantl ⊢ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V )
20 11 12 19 3jaoi ⊢ ( ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V )
21 4 20 sylbi ⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V )
22 onintrab ⊢ ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ V ↔ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On )
23 21 22 sylib ⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On )
24 23 adantl ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On )
25 onelon ⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → 𝑦 ∈ On )
26 25 expcom ⊢ ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On → 𝑦 ∈ On ) )
27 24 26 syl5 ⊢ ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → 𝑦 ∈ On ) )
28 fveq2 ⊢ ( 𝑎 = 𝑦 → ( 𝐴 ‘ 𝑎 ) = ( 𝐴 ‘ 𝑦 ) )
29 fveq2 ⊢ ( 𝑎 = 𝑦 → ( 𝐵 ‘ 𝑎 ) = ( 𝐵 ‘ 𝑦 ) )
30 28 29 neeq12d ⊢ ( 𝑎 = 𝑦 → ( ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) ↔ ( 𝐴 ‘ 𝑦 ) ≠ ( 𝐵 ‘ 𝑦 ) ) )
31 30 onnminsb ⊢ ( 𝑦 ∈ On → ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ¬ ( 𝐴 ‘ 𝑦 ) ≠ ( 𝐵 ‘ 𝑦 ) ) )
32 31 com12 ⊢ ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝑦 ∈ On → ¬ ( 𝐴 ‘ 𝑦 ) ≠ ( 𝐵 ‘ 𝑦 ) ) )
33 27 32 syldc ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ¬ ( 𝐴 ‘ 𝑦 ) ≠ ( 𝐵 ‘ 𝑦 ) ) )
34 df-ne ⊢ ( ( 𝐴 ‘ 𝑦 ) ≠ ( 𝐵 ‘ 𝑦 ) ↔ ¬ ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) )
35 34 con2bii ⊢ ( ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ↔ ¬ ( 𝐴 ‘ 𝑦 ) ≠ ( 𝐵 ‘ 𝑦 ) )
36 33 35 imbitrrdi ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) )
37 36 ralrimiv ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) )
38 24 37 jca ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) )
39 38 ex ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ) )
40 39 impac ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → ( ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
41 anass ⊢ ( ( ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ↔ ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) )
42 40 41 sylib ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) )
43 raleq ⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) )
44 fveq2 ⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
45 fveq2 ⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
46 44 45 breq12d ⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
47 43 46 anbi12d ⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) )
48 47 rspcev ⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) )
49 42 48 syl ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) )
50 49 ex ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) ) )
51 eqeq12 ⊢ ( ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = ∅ ) → ( ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) ↔ 1o = ∅ ) )
52 6 51 mtbiri ⊢ ( ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = ∅ ) → ¬ ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) )
53 1on ⊢ 1o ∈ On
54 0elon ⊢ ∅ ∈ On
55 suc11 ⊢ ( ( 1o ∈ On ∧ ∅ ∈ On ) → ( suc 1o = suc ∅ ↔ 1o = ∅ ) )
56 55 necon3bid ⊢ ( ( 1o ∈ On ∧ ∅ ∈ On ) → ( suc 1o ≠ suc ∅ ↔ 1o ≠ ∅ ) )
57 53 54 56 mp2an ⊢ ( suc 1o ≠ suc ∅ ↔ 1o ≠ ∅ )
58 5 57 mpbir ⊢ suc 1o ≠ suc ∅
59 df-2o ⊢ 2o = suc 1o
60 df-1o ⊢ 1o = suc ∅
61 59 60 eqeq12i ⊢ ( 2o = 1o ↔ suc 1o = suc ∅ )
62 58 61 nemtbir ⊢ ¬ 2o = 1o
63 eqeq12 ⊢ ( ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) → ( ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) ↔ 1o = 2o ) )
64 eqcom ⊢ ( 1o = 2o ↔ 2o = 1o )
65 63 64 bitrdi ⊢ ( ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) → ( ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) ↔ 2o = 1o ) )
66 62 65 mtbiri ⊢ ( ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) → ¬ ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) )
67 13 nesymi ⊢ ¬ ∅ = 2o
68 eqeq12 ⊢ ( ( ( 𝐴 ‘ 𝑥 ) = ∅ ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) → ( ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) ↔ ∅ = 2o ) )
69 67 68 mtbiri ⊢ ( ( ( 𝐴 ‘ 𝑥 ) = ∅ ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) → ¬ ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) )
70 52 66 69 3jaoi ⊢ ( ( ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑥 ) = ∅ ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) ) → ¬ ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) )
71 fvex ⊢ ( 𝐴 ‘ 𝑥 ) ∈ V
72 fvex ⊢ ( 𝐵 ‘ 𝑥 ) ∈ V
73 71 72 brtp ⊢ ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ↔ ( ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑥 ) = 1o ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑥 ) = ∅ ∧ ( 𝐵 ‘ 𝑥 ) = 2o ) ) )
74 df-ne ⊢ ( ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) ↔ ¬ ( 𝐴 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) )
75 70 73 74 3imtr4i ⊢ ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) → ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) )
76 fveq2 ⊢ ( 𝑎 = 𝑥 → ( 𝐴 ‘ 𝑎 ) = ( 𝐴 ‘ 𝑥 ) )
77 fveq2 ⊢ ( 𝑎 = 𝑥 → ( 𝐵 ‘ 𝑎 ) = ( 𝐵 ‘ 𝑥 ) )
78 76 77 neeq12d ⊢ ( 𝑎 = 𝑥 → ( ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) ↔ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) ) )
79 78 elrab ⊢ ( 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ↔ ( 𝑥 ∈ On ∧ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) ) )
80 79 biimpri ⊢ ( ( 𝑥 ∈ On ∧ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) ) → 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
81 80 adantlr ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) ) → 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
82 ssrab2 ⊢ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ On
83 ne0i ⊢ ( 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ≠ ∅ )
84 83 adantl ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ≠ ∅ )
85 onint ⊢ ( ( { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ On ∧ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ≠ ∅ ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
86 82 84 85 sylancr ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
87 nfrab1 ⊢ Ⅎ 𝑎 { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) }
88 87 nfint ⊢ Ⅎ 𝑎 ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) }
89 nfcv ⊢ Ⅎ 𝑎 On
90 nfcv ⊢ Ⅎ 𝑎 𝐴
91 90 88 nffv ⊢ Ⅎ 𝑎 ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
92 nfcv ⊢ Ⅎ 𝑎 𝐵
93 92 88 nffv ⊢ Ⅎ 𝑎 ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
94 91 93 nfne ⊢ Ⅎ 𝑎 ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ≠ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
95 fveq2 ⊢ ( 𝑎 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐴 ‘ 𝑎 ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
96 fveq2 ⊢ ( 𝑎 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐵 ‘ 𝑎 ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
97 95 96 neeq12d ⊢ ( 𝑎 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ≠ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
98 88 89 94 97 elrabf ⊢ ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ↔ ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ≠ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
99 98 simprbi ⊢ ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ≠ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
100 86 99 syl ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ≠ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
101 df-ne ⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ≠ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ↔ ¬ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
102 100 101 sylib ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ¬ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
103 fveq2 ⊢ ( 𝑦 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐴 ‘ 𝑦 ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
104 fveq2 ⊢ ( 𝑦 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐵 ‘ 𝑦 ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
105 103 104 eqeq12d ⊢ ( 𝑦 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
106 105 rspccv ⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ 𝑥 → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
107 106 ad2antlr ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ 𝑥 → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
108 102 107 mtod ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ¬ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ 𝑥 )
109 simpll ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → 𝑥 ∈ On )
110 oninton ⊢ ( ( { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ On ∧ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ≠ ∅ ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On )
111 82 83 110 sylancr ⊢ ( 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On )
112 111 adantl ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On )
113 ontri1 ⊢ ( ( 𝑥 ∈ On ∧ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ) → ( 𝑥 ⊆ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ↔ ¬ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ 𝑥 ) )
114 109 112 113 syl2anc ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( 𝑥 ⊆ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ↔ ¬ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ 𝑥 ) )
115 108 114 mpbird ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → 𝑥 ⊆ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
116 intss1 ⊢ ( 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ 𝑥 )
117 116 adantl ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ 𝑥 )
118 115 117 eqssd ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ 𝑥 ∈ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
119 81 118 syldan ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) ) → 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
120 75 119 sylan2 ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) → 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } )
121 120 fveq2d ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
122 120 fveq2d ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
123 121 122 breq12d ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) → ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
124 123 biimpd ⊢ ( ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) → ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
125 124 ex ⊢ ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) → ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) → ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) )
126 125 pm2.43d ⊢ ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) → ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
127 126 expimpd ⊢ ( 𝑥 ∈ On → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )
128 127 rexlimiv ⊢ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) )
129 50 128 impbid1 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) ) )
130 1 129 bitr4d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) )