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Theorem domunsncan 8101
 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 3810 . . . 4 𝑌 ⊆ ({𝐵} ∪ 𝑌)
2 reldom 8003 . . . . . 6 Rel ≼
32brrelex2i 5193 . . . . 5 (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ({𝐵} ∪ 𝑌) ∈ V)
43adantl 481 . . . 4 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → ({𝐵} ∪ 𝑌) ∈ V)
5 ssexg 4837 . . . 4 ((𝑌 ⊆ ({𝐵} ∪ 𝑌) ∧ ({𝐵} ∪ 𝑌) ∈ V) → 𝑌 ∈ V)
61, 4, 5sylancr 696 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑌 ∈ V)
7 brdomi 8008 . . . . 5 (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
8 vex 3234 . . . . . . . . . . 11 𝑓 ∈ V
98resex 5478 . . . . . . . . . 10 (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∈ V
10 simprr 811 . . . . . . . . . . 11 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
11 difss 3770 . . . . . . . . . . 11 (({𝐴} ∪ 𝑋) ∖ {𝐴}) ⊆ ({𝐴} ∪ 𝑋)
12 f1ores 6189 . . . . . . . . . . 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 695 . . . . . . . . . 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 8013 . . . . . . . . . 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 696 . . . . . . . . 9 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})))
16 df-f1 5931 . . . . . . . . . . . . 13 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) ↔ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ Fun 𝑓))
1716simprbi 479 . . . . . . . . . . . 12 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → Fun 𝑓)
18 imadif 6011 . . . . . . . . . . . 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 765 . . . . . . . . . 10 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
21 snex 4938 . . . . . . . . . . . . . 14 {𝐵} ∈ V
22 simprl 809 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑌 ∈ V)
23 unexg 7001 . . . . . . . . . . . . . 14 (({𝐵} ∈ V ∧ 𝑌 ∈ V) → ({𝐵} ∪ 𝑌) ∈ V)
2421, 22, 23sylancr 696 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ({𝐵} ∪ 𝑌) ∈ V)
25 difexg 4841 . . . . . . . . . . . . 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 6139 . . . . . . . . . . . . . . . 16 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌))
28 imassrn 5512 . . . . . . . . . . . . . . . . 17 (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ran 𝑓
29 frn 6091 . . . . . . . . . . . . . . . . 17 (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) → ran 𝑓 ⊆ ({𝐵} ∪ 𝑌))
3028, 29syl5ss 3647 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
3332ssdifd 3779 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
34 f1fn 6140 . . . . . . . . . . . . . . . 16 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓 Fn ({𝐴} ∪ 𝑋))
3534ad2antll 765 . . . . . . . . . . . . . . 15 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓 Fn ({𝐴} ∪ 𝑋))
36 domunsncan.a . . . . . . . . . . . . . . . . 17 𝐴 ∈ V
3736snid 4241 . . . . . . . . . . . . . . . 16 𝐴 ∈ {𝐴}
38 elun1 3813 . . . . . . . . . . . . . . . 16 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝑋))
3937, 38ax-mp 5 . . . . . . . . . . . . . . 15 𝐴 ∈ ({𝐴} ∪ 𝑋)
40 fnsnfv 6297 . . . . . . . . . . . . . . 15 ((𝑓 Fn ({𝐴} ∪ 𝑋) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
4135, 39, 40sylancl 695 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
4241difeq2d 3761 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) = (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
4333, 42sseqtr4d 3675 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}))
44 ssdomg 8043 . . . . . . . . . . . 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 6397 . . . . . . . . . . . . . 14 ((𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
4727, 39, 46sylancl 695 . . . . . . . . . . . . 13 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
4847ad2antll 765 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
49 domunsncan.b . . . . . . . . . . . . . 14 𝐵 ∈ V
5049snid 4241 . . . . . . . . . . . . 13 𝐵 ∈ {𝐵}
51 elun1 3813 . . . . . . . . . . . . 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 8083 . . . . . . . . . . . 12 ((({𝐵} ∪ 𝑌) ∈ V ∧ (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌) ∧ 𝐵 ∈ ({𝐵} ∪ 𝑌)) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5424, 48, 52, 53syl3anc 1366 . . . . . . . . . . 11 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
55 domentr 8056 . . . . . . . . . . 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 694 . . . . . . . . . 10 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5720, 56eqbrtrd 4707 . . . . . . . . 9 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
58 endomtr 8055 . . . . . . . . 9 (((({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∧ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5915, 57, 58syl2anc 694 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
60 uncom 3790 . . . . . . . . . . . 12 ({𝐴} ∪ 𝑋) = (𝑋 ∪ {𝐴})
6160difeq1i 3757 . . . . . . . . . . 11 (({𝐴} ∪ 𝑋) ∖ {𝐴}) = ((𝑋 ∪ {𝐴}) ∖ {𝐴})
62 difun2 4081 . . . . . . . . . . 11 ((𝑋 ∪ {𝐴}) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
6361, 62eqtri 2673 . . . . . . . . . 10 (({𝐴} ∪ 𝑋) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
64 difsn 4360 . . . . . . . . . 10 𝐴𝑋 → (𝑋 ∖ {𝐴}) = 𝑋)
6563, 64syl5eq 2697 . . . . . . . . 9 𝐴𝑋 → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
6665ad2antrr 762 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
67 uncom 3790 . . . . . . . . . . . 12 ({𝐵} ∪ 𝑌) = (𝑌 ∪ {𝐵})
6867difeq1i 3757 . . . . . . . . . . 11 (({𝐵} ∪ 𝑌) ∖ {𝐵}) = ((𝑌 ∪ {𝐵}) ∖ {𝐵})
69 difun2 4081 . . . . . . . . . . 11 ((𝑌 ∪ {𝐵}) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
7068, 69eqtri 2673 . . . . . . . . . 10 (({𝐵} ∪ 𝑌) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
71 difsn 4360 . . . . . . . . . 10 𝐵𝑌 → (𝑌 ∖ {𝐵}) = 𝑌)
7270, 71syl5eq 2697 . . . . . . . . 9 𝐵𝑌 → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
7372ad2antlr 763 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
7459, 66, 733brtr3d 4716 . . . . . . 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 1901 . . . . 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 455 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → (𝑌 ∈ V → 𝑋𝑌))
796, 78mpd 15 . 2 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑋𝑌)
80 en2sn 8078 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴} ≈ {𝐵})
8136, 49, 80mp2an 708 . . . 4 {𝐴} ≈ {𝐵}
82 endom 8024 . . . 4 ({𝐴} ≈ {𝐵} → {𝐴} ≼ {𝐵})
8381, 82mp1i 13 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → {𝐴} ≼ {𝐵})
84 simpr 476 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → 𝑋𝑌)
85 incom 3838 . . . . 5 ({𝐵} ∩ 𝑌) = (𝑌 ∩ {𝐵})
86 disjsn 4278 . . . . . 6 ((𝑌 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝑌)
8786biimpri 218 . . . . 5 𝐵𝑌 → (𝑌 ∩ {𝐵}) = ∅)
8885, 87syl5eq 2697 . . . 4 𝐵𝑌 → ({𝐵} ∩ 𝑌) = ∅)
8988ad2antlr 763 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → ({𝐵} ∩ 𝑌) = ∅)
90 undom 8089 . . 3 ((({𝐴} ≼ {𝐵} ∧ 𝑋𝑌) ∧ ({𝐵} ∩ 𝑌) = ∅) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
9183, 84, 89, 90syl21anc 1365 . 2 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
9279, 91impbida 895 1 ((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210   class class class wbr 4685  ◡ccnv 5142  ran crn 5144   ↾ cres 5145   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  –1-1→wf1 5923  –1-1-onto→wf1o 5925  ‘cfv 5926   ≈ cen 7994   ≼ cdom 7995 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999 This theorem is referenced by:  domunfican  8274
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