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