Metamath Proof Explorer


Theorem unxpdom2

Description: Corollary of unxpdom . (Contributed by NM, 16-Sep-2004)

Ref Expression
Assertion unxpdom2 ( ( 1o𝐴𝐵𝐴 ) → ( 𝐴𝐵 ) ≼ ( 𝐴 × 𝐴 ) )

Proof

Step Hyp Ref Expression
1 relsdom Rel ≺
2 1 brrelex2i ( 1o𝐴𝐴 ∈ V )
3 2 adantr ( ( 1o𝐴𝐵𝐴 ) → 𝐴 ∈ V )
4 1onn 1o ∈ ω
5 xpsneng ( ( 𝐴 ∈ V ∧ 1o ∈ ω ) → ( 𝐴 × { 1o } ) ≈ 𝐴 )
6 3 4 5 sylancl ( ( 1o𝐴𝐵𝐴 ) → ( 𝐴 × { 1o } ) ≈ 𝐴 )
7 6 ensymd ( ( 1o𝐴𝐵𝐴 ) → 𝐴 ≈ ( 𝐴 × { 1o } ) )
8 endom ( 𝐴 ≈ ( 𝐴 × { 1o } ) → 𝐴 ≼ ( 𝐴 × { 1o } ) )
9 7 8 syl ( ( 1o𝐴𝐵𝐴 ) → 𝐴 ≼ ( 𝐴 × { 1o } ) )
10 simpr ( ( 1o𝐴𝐵𝐴 ) → 𝐵𝐴 )
11 0ex ∅ ∈ V
12 xpsneng ( ( 𝐴 ∈ V ∧ ∅ ∈ V ) → ( 𝐴 × { ∅ } ) ≈ 𝐴 )
13 3 11 12 sylancl ( ( 1o𝐴𝐵𝐴 ) → ( 𝐴 × { ∅ } ) ≈ 𝐴 )
14 13 ensymd ( ( 1o𝐴𝐵𝐴 ) → 𝐴 ≈ ( 𝐴 × { ∅ } ) )
15 domentr ( ( 𝐵𝐴𝐴 ≈ ( 𝐴 × { ∅ } ) ) → 𝐵 ≼ ( 𝐴 × { ∅ } ) )
16 10 14 15 syl2anc ( ( 1o𝐴𝐵𝐴 ) → 𝐵 ≼ ( 𝐴 × { ∅ } ) )
17 1n0 1o ≠ ∅
18 xpsndisj ( 1o ≠ ∅ → ( ( 𝐴 × { 1o } ) ∩ ( 𝐴 × { ∅ } ) ) = ∅ )
19 17 18 mp1i ( ( 1o𝐴𝐵𝐴 ) → ( ( 𝐴 × { 1o } ) ∩ ( 𝐴 × { ∅ } ) ) = ∅ )
20 undom ( ( ( 𝐴 ≼ ( 𝐴 × { 1o } ) ∧ 𝐵 ≼ ( 𝐴 × { ∅ } ) ) ∧ ( ( 𝐴 × { 1o } ) ∩ ( 𝐴 × { ∅ } ) ) = ∅ ) → ( 𝐴𝐵 ) ≼ ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) )
21 9 16 19 20 syl21anc ( ( 1o𝐴𝐵𝐴 ) → ( 𝐴𝐵 ) ≼ ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) )
22 sdomentr ( ( 1o𝐴𝐴 ≈ ( 𝐴 × { 1o } ) ) → 1o ≺ ( 𝐴 × { 1o } ) )
23 7 22 syldan ( ( 1o𝐴𝐵𝐴 ) → 1o ≺ ( 𝐴 × { 1o } ) )
24 sdomentr ( ( 1o𝐴𝐴 ≈ ( 𝐴 × { ∅ } ) ) → 1o ≺ ( 𝐴 × { ∅ } ) )
25 14 24 syldan ( ( 1o𝐴𝐵𝐴 ) → 1o ≺ ( 𝐴 × { ∅ } ) )
26 unxpdom ( ( 1o ≺ ( 𝐴 × { 1o } ) ∧ 1o ≺ ( 𝐴 × { ∅ } ) ) → ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) ≼ ( ( 𝐴 × { 1o } ) × ( 𝐴 × { ∅ } ) ) )
27 23 25 26 syl2anc ( ( 1o𝐴𝐵𝐴 ) → ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) ≼ ( ( 𝐴 × { 1o } ) × ( 𝐴 × { ∅ } ) ) )
28 xpen ( ( ( 𝐴 × { 1o } ) ≈ 𝐴 ∧ ( 𝐴 × { ∅ } ) ≈ 𝐴 ) → ( ( 𝐴 × { 1o } ) × ( 𝐴 × { ∅ } ) ) ≈ ( 𝐴 × 𝐴 ) )
29 6 13 28 syl2anc ( ( 1o𝐴𝐵𝐴 ) → ( ( 𝐴 × { 1o } ) × ( 𝐴 × { ∅ } ) ) ≈ ( 𝐴 × 𝐴 ) )
30 domentr ( ( ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) ≼ ( ( 𝐴 × { 1o } ) × ( 𝐴 × { ∅ } ) ) ∧ ( ( 𝐴 × { 1o } ) × ( 𝐴 × { ∅ } ) ) ≈ ( 𝐴 × 𝐴 ) ) → ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) ≼ ( 𝐴 × 𝐴 ) )
31 27 29 30 syl2anc ( ( 1o𝐴𝐵𝐴 ) → ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) ≼ ( 𝐴 × 𝐴 ) )
32 domtr ( ( ( 𝐴𝐵 ) ≼ ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) ∧ ( ( 𝐴 × { 1o } ) ∪ ( 𝐴 × { ∅ } ) ) ≼ ( 𝐴 × 𝐴 ) ) → ( 𝐴𝐵 ) ≼ ( 𝐴 × 𝐴 ) )
33 21 31 32 syl2anc ( ( 1o𝐴𝐵𝐴 ) → ( 𝐴𝐵 ) ≼ ( 𝐴 × 𝐴 ) )