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Theorem domunsncan 8007
Description: A singleton cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
Hypotheses
Ref Expression
domunsncan.a 𝐴 ∈ V
domunsncan.b 𝐵 ∈ V
Assertion
Ref Expression
domunsncan ((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋𝑌))

Proof of Theorem domunsncan
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ssun2 3757 . . . 4 𝑌 ⊆ ({𝐵} ∪ 𝑌)
2 reldom 7908 . . . . . 6 Rel ≼
32brrelex2i 5121 . . . . 5 (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ({𝐵} ∪ 𝑌) ∈ V)
43adantl 482 . . . 4 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → ({𝐵} ∪ 𝑌) ∈ V)
5 ssexg 4766 . . . 4 ((𝑌 ⊆ ({𝐵} ∪ 𝑌) ∧ ({𝐵} ∪ 𝑌) ∈ V) → 𝑌 ∈ V)
61, 4, 5sylancr 694 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑌 ∈ V)
7 brdomi 7913 . . . . 5 (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
8 vex 3189 . . . . . . . . . . 11 𝑓 ∈ V
98resex 5404 . . . . . . . . . 10 (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∈ V
10 simprr 795 . . . . . . . . . . 11 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
11 difss 3717 . . . . . . . . . . 11 (({𝐴} ∪ 𝑋) ∖ {𝐴}) ⊆ ({𝐴} ∪ 𝑋)
12 f1ores 6110 . . . . . . . . . . 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})))
1310, 11, 12sylancl 693 . . . . . . . . . 10 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (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 7918 . . . . . . . . . 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})))
159, 13, 14sylancr 694 . . . . . . . . 9 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})))
16 df-f1 5854 . . . . . . . . . . . . 13 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) ↔ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ Fun 𝑓))
1716simprbi 480 . . . . . . . . . . . 12 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → Fun 𝑓)
18 imadif 5933 . . . . . . . . . . . 12 (Fun 𝑓 → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
1917, 18syl 17 . . . . . . . . . . 11 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
2019ad2antll 764 . . . . . . . . . 10 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
21 snex 4871 . . . . . . . . . . . . . 14 {𝐵} ∈ V
22 simprl 793 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑌 ∈ V)
23 unexg 6915 . . . . . . . . . . . . . 14 (({𝐵} ∈ V ∧ 𝑌 ∈ V) → ({𝐵} ∪ 𝑌) ∈ V)
2421, 22, 23sylancr 694 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ({𝐵} ∪ 𝑌) ∈ V)
25 difexg 4770 . . . . . . . . . . . . 13 (({𝐵} ∪ 𝑌) ∈ V → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ∈ V)
2624, 25syl 17 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ∈ V)
27 f1f 6060 . . . . . . . . . . . . . . . 16 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌))
28 imassrn 5438 . . . . . . . . . . . . . . . . 17 (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ran 𝑓
29 frn 6012 . . . . . . . . . . . . . . . . 17 (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) → ran 𝑓 ⊆ ({𝐵} ∪ 𝑌))
3028, 29syl5ss 3595 . . . . . . . . . . . . . . . 16 (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
3127, 30syl 17 . . . . . . . . . . . . . . 15 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
3231ad2antll 764 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
3332ssdifd 3726 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
34 f1fn 6061 . . . . . . . . . . . . . . . 16 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓 Fn ({𝐴} ∪ 𝑋))
3534ad2antll 764 . . . . . . . . . . . . . . 15 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓 Fn ({𝐴} ∪ 𝑋))
36 domunsncan.a . . . . . . . . . . . . . . . . 17 𝐴 ∈ V
3736snid 4181 . . . . . . . . . . . . . . . 16 𝐴 ∈ {𝐴}
38 elun1 3760 . . . . . . . . . . . . . . . 16 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝑋))
3937, 38ax-mp 5 . . . . . . . . . . . . . . 15 𝐴 ∈ ({𝐴} ∪ 𝑋)
40 fnsnfv 6217 . . . . . . . . . . . . . . 15 ((𝑓 Fn ({𝐴} ∪ 𝑋) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
4135, 39, 40sylancl 693 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
4241difeq2d 3708 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) = (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
4333, 42sseqtr4d 3623 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}))
44 ssdomg 7948 . . . . . . . . . . . 12 ((({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ∈ V → (((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)})))
4526, 43, 44sylc 65 . . . . . . . . . . 11 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}))
46 ffvelrn 6315 . . . . . . . . . . . . . 14 ((𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
4727, 39, 46sylancl 693 . . . . . . . . . . . . 13 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
4847ad2antll 764 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
49 domunsncan.b . . . . . . . . . . . . . 14 𝐵 ∈ V
5049snid 4181 . . . . . . . . . . . . 13 𝐵 ∈ {𝐵}
51 elun1 3760 . . . . . . . . . . . . 13 (𝐵 ∈ {𝐵} → 𝐵 ∈ ({𝐵} ∪ 𝑌))
5250, 51mp1i 13 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝐵 ∈ ({𝐵} ∪ 𝑌))
53 difsnen 7989 . . . . . . . . . . . 12 ((({𝐵} ∪ 𝑌) ∈ V ∧ (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌) ∧ 𝐵 ∈ ({𝐵} ∪ 𝑌)) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5424, 48, 52, 53syl3anc 1323 . . . . . . . . . . 11 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
55 domentr 7962 . . . . . . . . . . 11 ((((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ∧ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5645, 54, 55syl2anc 692 . . . . . . . . . 10 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5720, 56eqbrtrd 4637 . . . . . . . . 9 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
58 endomtr 7961 . . . . . . . . 9 (((({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∧ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5915, 57, 58syl2anc 692 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
60 uncom 3737 . . . . . . . . . . . 12 ({𝐴} ∪ 𝑋) = (𝑋 ∪ {𝐴})
6160difeq1i 3704 . . . . . . . . . . 11 (({𝐴} ∪ 𝑋) ∖ {𝐴}) = ((𝑋 ∪ {𝐴}) ∖ {𝐴})
62 difun2 4022 . . . . . . . . . . 11 ((𝑋 ∪ {𝐴}) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
6361, 62eqtri 2643 . . . . . . . . . 10 (({𝐴} ∪ 𝑋) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
64 difsn 4299 . . . . . . . . . 10 𝐴𝑋 → (𝑋 ∖ {𝐴}) = 𝑋)
6563, 64syl5eq 2667 . . . . . . . . 9 𝐴𝑋 → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
6665ad2antrr 761 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
67 uncom 3737 . . . . . . . . . . . 12 ({𝐵} ∪ 𝑌) = (𝑌 ∪ {𝐵})
6867difeq1i 3704 . . . . . . . . . . 11 (({𝐵} ∪ 𝑌) ∖ {𝐵}) = ((𝑌 ∪ {𝐵}) ∖ {𝐵})
69 difun2 4022 . . . . . . . . . . 11 ((𝑌 ∪ {𝐵}) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
7068, 69eqtri 2643 . . . . . . . . . 10 (({𝐵} ∪ 𝑌) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
71 difsn 4299 . . . . . . . . . 10 𝐵𝑌 → (𝑌 ∖ {𝐵}) = 𝑌)
7270, 71syl5eq 2667 . . . . . . . . 9 𝐵𝑌 → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
7372ad2antlr 762 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
7459, 66, 733brtr3d 4646 . . . . . . 7 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑋𝑌)
7574expr 642 . . . . . 6 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑌 ∈ V) → (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋𝑌))
7675exlimdv 1858 . . . . 5 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑌 ∈ V) → (∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋𝑌))
777, 76syl5 34 . . . 4 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑌 ∈ V) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → 𝑋𝑌))
7877impancom 456 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → (𝑌 ∈ V → 𝑋𝑌))
796, 78mpd 15 . 2 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑋𝑌)
80 en2sn 7984 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴} ≈ {𝐵})
8136, 49, 80mp2an 707 . . . 4 {𝐴} ≈ {𝐵}
82 endom 7929 . . . 4 ({𝐴} ≈ {𝐵} → {𝐴} ≼ {𝐵})
8381, 82mp1i 13 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → {𝐴} ≼ {𝐵})
84 simpr 477 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → 𝑋𝑌)
85 incom 3785 . . . . 5 ({𝐵} ∩ 𝑌) = (𝑌 ∩ {𝐵})
86 disjsn 4218 . . . . . 6 ((𝑌 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝑌)
8786biimpri 218 . . . . 5 𝐵𝑌 → (𝑌 ∩ {𝐵}) = ∅)
8885, 87syl5eq 2667 . . . 4 𝐵𝑌 → ({𝐵} ∩ 𝑌) = ∅)
8988ad2antlr 762 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → ({𝐵} ∩ 𝑌) = ∅)
90 undom 7995 . . 3 ((({𝐴} ≼ {𝐵} ∧ 𝑋𝑌) ∧ ({𝐵} ∩ 𝑌) = ∅) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
9183, 84, 89, 90syl21anc 1322 . 2 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
9279, 91impbida 876 1 ((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cdif 3553  cun 3554  cin 3555  wss 3556  c0 3893  {csn 4150   class class class wbr 4615  ccnv 5075  ran crn 5077  cres 5078  cima 5079  Fun wfun 5843   Fn wfn 5844  wf 5845  1-1wf1 5846  1-1-ontowf1o 5848  cfv 5849  cen 7899  cdom 7900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-1o 7508  df-er 7690  df-en 7903  df-dom 7904
This theorem is referenced by:  domunfican  8180
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