Step | Hyp | Ref
| Expression |
1 | | ssun2 4148 |
. . . 4
⊢ 𝑌 ⊆ ({𝐵} ∪ 𝑌) |
2 | | reldom 8514 |
. . . . . 6
⊢ Rel
≼ |
3 | 2 | brrelex2i 5608 |
. . . . 5
⊢ (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ({𝐵} ∪ 𝑌) ∈ V) |
4 | 3 | adantl 484 |
. . . 4
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → ({𝐵} ∪ 𝑌) ∈ V) |
5 | | ssexg 5226 |
. . . 4
⊢ ((𝑌 ⊆ ({𝐵} ∪ 𝑌) ∧ ({𝐵} ∪ 𝑌) ∈ V) → 𝑌 ∈ V) |
6 | 1, 4, 5 | sylancr 589 |
. . 3
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑌 ∈ V) |
7 | | brdomi 8519 |
. . . . 5
⊢ (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌)) |
8 | | vex 3497 |
. . . . . . . . . . 11
⊢ 𝑓 ∈ V |
9 | 8 | resex 5898 |
. . . . . . . . . 10
⊢ (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∈ V |
10 | | simprr 771 |
. . . . . . . . . . 11
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌)) |
11 | | difss 4107 |
. . . . . . . . . . 11
⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) ⊆ ({𝐴} ∪ 𝑋) |
12 | | f1ores 6628 |
. . . . . . . . . . 11
⊢ ((𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) ∧ (({𝐴} ∪ 𝑋) ∖ {𝐴}) ⊆ ({𝐴} ∪ 𝑋)) → (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})):(({𝐴} ∪ 𝑋) ∖ {𝐴})–1-1-onto→(𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴}))) |
13 | 10, 11, 12 | sylancl 588 |
. . . . . . . . . 10
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})):(({𝐴} ∪ 𝑋) ∖ {𝐴})–1-1-onto→(𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴}))) |
14 | | f1oen3g 8524 |
. . . . . . . . . 10
⊢ (((𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∈ V ∧ (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})):(({𝐴} ∪ 𝑋) ∖ {𝐴})–1-1-onto→(𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴}))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴}))) |
15 | 9, 13, 14 | sylancr 589 |
. . . . . . . . 9
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴}))) |
16 | | df-f1 6359 |
. . . . . . . . . . . 12
⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) ↔ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ Fun ◡𝑓)) |
17 | | imadif 6437 |
. . . . . . . . . . . 12
⊢ (Fun
◡𝑓 → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴}))) |
18 | 16, 17 | simplbiim 507 |
. . . . . . . . . . 11
⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴}))) |
19 | 18 | ad2antll 727 |
. . . . . . . . . 10
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴}))) |
20 | | snex 5331 |
. . . . . . . . . . . . . 14
⊢ {𝐵} ∈ V |
21 | | simprl 769 |
. . . . . . . . . . . . . 14
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑌 ∈ V) |
22 | | unexg 7471 |
. . . . . . . . . . . . . 14
⊢ (({𝐵} ∈ V ∧ 𝑌 ∈ V) → ({𝐵} ∪ 𝑌) ∈ V) |
23 | 20, 21, 22 | sylancr 589 |
. . . . . . . . . . . . 13
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ({𝐵} ∪ 𝑌) ∈ V) |
24 | | difexg 5230 |
. . . . . . . . . . . . 13
⊢ (({𝐵} ∪ 𝑌) ∈ V → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∈ V) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . 12
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∈ V) |
26 | | f1f 6574 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌)) |
27 | | fimass 6554 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌)) |
28 | 26, 27 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌)) |
29 | 28 | ad2antll 727 |
. . . . . . . . . . . . . 14
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌)) |
30 | 29 | ssdifd 4116 |
. . . . . . . . . . . . 13
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴}))) |
31 | | f1fn 6575 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓 Fn ({𝐴} ∪ 𝑋)) |
32 | 31 | ad2antll 727 |
. . . . . . . . . . . . . . 15
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓 Fn ({𝐴} ∪ 𝑋)) |
33 | | domunsncan.a |
. . . . . . . . . . . . . . . . 17
⊢ 𝐴 ∈ V |
34 | 33 | snid 4600 |
. . . . . . . . . . . . . . . 16
⊢ 𝐴 ∈ {𝐴} |
35 | | elun1 4151 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝑋)) |
36 | 34, 35 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ 𝐴 ∈ ({𝐴} ∪ 𝑋) |
37 | | fnsnfv 6742 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 Fn ({𝐴} ∪ 𝑋) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴})) |
38 | 32, 36, 37 | sylancl 588 |
. . . . . . . . . . . . . 14
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴})) |
39 | 38 | difeq2d 4098 |
. . . . . . . . . . . . 13
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) = (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴}))) |
40 | 30, 39 | sseqtrrd 4007 |
. . . . . . . . . . . 12
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)})) |
41 | | ssdomg 8554 |
. . . . . . . . . . . 12
⊢ ((({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∈ V → (((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}))) |
42 | 25, 40, 41 | sylc 65 |
. . . . . . . . . . 11
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)})) |
43 | | ffvelrn 6848 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌)) |
44 | 26, 36, 43 | sylancl 588 |
. . . . . . . . . . . . 13
⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌)) |
45 | 44 | ad2antll 727 |
. . . . . . . . . . . 12
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌)) |
46 | | domunsncan.b |
. . . . . . . . . . . . . 14
⊢ 𝐵 ∈ V |
47 | 46 | snid 4600 |
. . . . . . . . . . . . 13
⊢ 𝐵 ∈ {𝐵} |
48 | | elun1 4151 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ ({𝐵} ∪ 𝑌)) |
49 | 47, 48 | mp1i 13 |
. . . . . . . . . . . 12
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝐵 ∈ ({𝐵} ∪ 𝑌)) |
50 | | difsnen 8598 |
. . . . . . . . . . . 12
⊢ ((({𝐵} ∪ 𝑌) ∈ V ∧ (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌) ∧ 𝐵 ∈ ({𝐵} ∪ 𝑌)) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵})) |
51 | 23, 45, 49, 50 | syl3anc 1367 |
. . . . . . . . . . 11
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵})) |
52 | | domentr 8567 |
. . . . . . . . . . 11
⊢ ((((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∧ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) |
53 | 42, 51, 52 | syl2anc 586 |
. . . . . . . . . 10
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) |
54 | 19, 53 | eqbrtrd 5087 |
. . . . . . . . 9
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) |
55 | | endomtr 8566 |
. . . . . . . . 9
⊢
(((({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∧ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) |
56 | 15, 54, 55 | syl2anc 586 |
. . . . . . . 8
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) |
57 | | uncom 4128 |
. . . . . . . . . . . 12
⊢ ({𝐴} ∪ 𝑋) = (𝑋 ∪ {𝐴}) |
58 | 57 | difeq1i 4094 |
. . . . . . . . . . 11
⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) = ((𝑋 ∪ {𝐴}) ∖ {𝐴}) |
59 | | difun2 4428 |
. . . . . . . . . . 11
⊢ ((𝑋 ∪ {𝐴}) ∖ {𝐴}) = (𝑋 ∖ {𝐴}) |
60 | 58, 59 | eqtri 2844 |
. . . . . . . . . 10
⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) = (𝑋 ∖ {𝐴}) |
61 | | difsn 4730 |
. . . . . . . . . 10
⊢ (¬
𝐴 ∈ 𝑋 → (𝑋 ∖ {𝐴}) = 𝑋) |
62 | 60, 61 | syl5eq 2868 |
. . . . . . . . 9
⊢ (¬
𝐴 ∈ 𝑋 → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋) |
63 | 62 | ad2antrr 724 |
. . . . . . . 8
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋) |
64 | | uncom 4128 |
. . . . . . . . . . . 12
⊢ ({𝐵} ∪ 𝑌) = (𝑌 ∪ {𝐵}) |
65 | 64 | difeq1i 4094 |
. . . . . . . . . . 11
⊢ (({𝐵} ∪ 𝑌) ∖ {𝐵}) = ((𝑌 ∪ {𝐵}) ∖ {𝐵}) |
66 | | difun2 4428 |
. . . . . . . . . . 11
⊢ ((𝑌 ∪ {𝐵}) ∖ {𝐵}) = (𝑌 ∖ {𝐵}) |
67 | 65, 66 | eqtri 2844 |
. . . . . . . . . 10
⊢ (({𝐵} ∪ 𝑌) ∖ {𝐵}) = (𝑌 ∖ {𝐵}) |
68 | | difsn 4730 |
. . . . . . . . . 10
⊢ (¬
𝐵 ∈ 𝑌 → (𝑌 ∖ {𝐵}) = 𝑌) |
69 | 67, 68 | syl5eq 2868 |
. . . . . . . . 9
⊢ (¬
𝐵 ∈ 𝑌 → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌) |
70 | 69 | ad2antlr 725 |
. . . . . . . 8
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌) |
71 | 56, 63, 70 | 3brtr3d 5096 |
. . . . . . 7
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑋 ≼ 𝑌) |
72 | 71 | expr 459 |
. . . . . 6
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌)) |
73 | 72 | exlimdv 1930 |
. . . . 5
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌)) |
74 | 7, 73 | syl5 34 |
. . . 4
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌)) |
75 | 74 | impancom 454 |
. . 3
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → (𝑌 ∈ V → 𝑋 ≼ 𝑌)) |
76 | 6, 75 | mpd 15 |
. 2
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑋 ≼ 𝑌) |
77 | | en2sn 8592 |
. . . . 5
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴} ≈ {𝐵}) |
78 | 33, 46, 77 | mp2an 690 |
. . . 4
⊢ {𝐴} ≈ {𝐵} |
79 | | endom 8535 |
. . . 4
⊢ ({𝐴} ≈ {𝐵} → {𝐴} ≼ {𝐵}) |
80 | 78, 79 | mp1i 13 |
. . 3
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → {𝐴} ≼ {𝐵}) |
81 | | simpr 487 |
. . 3
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → 𝑋 ≼ 𝑌) |
82 | | incom 4177 |
. . . . 5
⊢ ({𝐵} ∩ 𝑌) = (𝑌 ∩ {𝐵}) |
83 | | disjsn 4646 |
. . . . . 6
⊢ ((𝑌 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝑌) |
84 | 83 | biimpri 230 |
. . . . 5
⊢ (¬
𝐵 ∈ 𝑌 → (𝑌 ∩ {𝐵}) = ∅) |
85 | 82, 84 | syl5eq 2868 |
. . . 4
⊢ (¬
𝐵 ∈ 𝑌 → ({𝐵} ∩ 𝑌) = ∅) |
86 | 85 | ad2antlr 725 |
. . 3
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → ({𝐵} ∩ 𝑌) = ∅) |
87 | | undom 8604 |
. . 3
⊢ ((({𝐴} ≼ {𝐵} ∧ 𝑋 ≼ 𝑌) ∧ ({𝐵} ∩ 𝑌) = ∅) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) |
88 | 80, 81, 86, 87 | syl21anc 835 |
. 2
⊢ (((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) |
89 | 76, 88 | impbida 799 |
1
⊢ ((¬
𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋 ≼ 𝑌)) |