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Theorem domunsncan 9075
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 4174 . . . 4 𝑌 ⊆ ({𝐵} ∪ 𝑌)
2 reldom 8948 . . . . . 6 Rel ≼
32brrelex2i 5734 . . . . 5 (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ({𝐵} ∪ 𝑌) ∈ V)
43adantl 481 . . . 4 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → ({𝐵} ∪ 𝑌) ∈ V)
5 ssexg 5324 . . . 4 ((𝑌 ⊆ ({𝐵} ∪ 𝑌) ∧ ({𝐵} ∪ 𝑌) ∈ V) → 𝑌 ∈ V)
61, 4, 5sylancr 586 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑌 ∈ V)
7 brdomi 8957 . . . . 5 (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
8 vex 3477 . . . . . . . . . . 11 𝑓 ∈ V
98resex 6030 . . . . . . . . . 10 (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∈ V
10 simprr 770 . . . . . . . . . . 11 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
11 difss 4132 . . . . . . . . . . 11 (({𝐴} ∪ 𝑋) ∖ {𝐴}) ⊆ ({𝐴} ∪ 𝑋)
12 f1ores 6848 . . . . . . . . . . 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 585 . . . . . . . . . 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 8965 . . . . . . . . . 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 586 . . . . . . . . 9 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})))
16 df-f1 6549 . . . . . . . . . . . 12 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) ↔ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ Fun 𝑓))
17 imadif 6633 . . . . . . . . . . . 12 (Fun 𝑓 → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
1816, 17simplbiim 504 . . . . . . . . . . 11 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
1918ad2antll 726 . . . . . . . . . 10 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
20 snex 5432 . . . . . . . . . . . . . 14 {𝐵} ∈ V
21 simprl 768 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑌 ∈ V)
22 unexg 7739 . . . . . . . . . . . . . 14 (({𝐵} ∈ V ∧ 𝑌 ∈ V) → ({𝐵} ∪ 𝑌) ∈ V)
2320, 21, 22sylancr 586 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ({𝐵} ∪ 𝑌) ∈ V)
2423difexd 5330 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ∈ V)
25 f1f 6788 . . . . . . . . . . . . . . . 16 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌))
26 fimass 6739 . . . . . . . . . . . . . . . 16 (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
2725, 26syl 17 . . . . . . . . . . . . . . 15 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
2827ad2antll 726 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
2928ssdifd 4141 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
30 f1fn 6789 . . . . . . . . . . . . . . . 16 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓 Fn ({𝐴} ∪ 𝑋))
3130ad2antll 726 . . . . . . . . . . . . . . 15 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓 Fn ({𝐴} ∪ 𝑋))
32 domunsncan.a . . . . . . . . . . . . . . . . 17 𝐴 ∈ V
3332snid 4665 . . . . . . . . . . . . . . . 16 𝐴 ∈ {𝐴}
34 elun1 4177 . . . . . . . . . . . . . . . 16 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝑋))
3533, 34ax-mp 5 . . . . . . . . . . . . . . 15 𝐴 ∈ ({𝐴} ∪ 𝑋)
36 fnsnfv 6971 . . . . . . . . . . . . . . 15 ((𝑓 Fn ({𝐴} ∪ 𝑋) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
3731, 35, 36sylancl 585 . . . . . . . . . . . . . 14 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
3837difeq2d 4123 . . . . . . . . . . . . 13 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) = (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
3929, 38sseqtrrd 4024 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}))
40 ssdomg 8999 . . . . . . . . . . . 12 ((({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ∈ V → (((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)})))
4124, 39, 40sylc 65 . . . . . . . . . . 11 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}))
42 ffvelcdm 7084 . . . . . . . . . . . . . 14 ((𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
4325, 35, 42sylancl 585 . . . . . . . . . . . . 13 (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
4443ad2antll 726 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌))
45 domunsncan.b . . . . . . . . . . . . . 14 𝐵 ∈ V
4645snid 4665 . . . . . . . . . . . . 13 𝐵 ∈ {𝐵}
47 elun1 4177 . . . . . . . . . . . . 13 (𝐵 ∈ {𝐵} → 𝐵 ∈ ({𝐵} ∪ 𝑌))
4846, 47mp1i 13 . . . . . . . . . . . 12 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝐵 ∈ ({𝐵} ∪ 𝑌))
49 difsnen 9056 . . . . . . . . . . . 12 ((({𝐵} ∪ 𝑌) ∈ V ∧ (𝑓𝐴) ∈ ({𝐵} ∪ 𝑌) ∧ 𝐵 ∈ ({𝐵} ∪ 𝑌)) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5023, 44, 48, 49syl3anc 1370 . . . . . . . . . . 11 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
51 domentr 9012 . . . . . . . . . . 11 ((((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ∧ (({𝐵} ∪ 𝑌) ∖ {(𝑓𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5241, 50, 51syl2anc 583 . . . . . . . . . 10 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5319, 52eqbrtrd 5171 . . . . . . . . 9 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
54 endomtr 9011 . . . . . . . . 9 (((({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∧ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
5515, 53, 54syl2anc 583 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
56 uncom 4154 . . . . . . . . . . . 12 ({𝐴} ∪ 𝑋) = (𝑋 ∪ {𝐴})
5756difeq1i 4119 . . . . . . . . . . 11 (({𝐴} ∪ 𝑋) ∖ {𝐴}) = ((𝑋 ∪ {𝐴}) ∖ {𝐴})
58 difun2 4481 . . . . . . . . . . 11 ((𝑋 ∪ {𝐴}) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
5957, 58eqtri 2759 . . . . . . . . . 10 (({𝐴} ∪ 𝑋) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
60 difsn 4802 . . . . . . . . . 10 𝐴𝑋 → (𝑋 ∖ {𝐴}) = 𝑋)
6159, 60eqtrid 2783 . . . . . . . . 9 𝐴𝑋 → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
6261ad2antrr 723 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
63 uncom 4154 . . . . . . . . . . . 12 ({𝐵} ∪ 𝑌) = (𝑌 ∪ {𝐵})
6463difeq1i 4119 . . . . . . . . . . 11 (({𝐵} ∪ 𝑌) ∖ {𝐵}) = ((𝑌 ∪ {𝐵}) ∖ {𝐵})
65 difun2 4481 . . . . . . . . . . 11 ((𝑌 ∪ {𝐵}) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
6664, 65eqtri 2759 . . . . . . . . . 10 (({𝐵} ∪ 𝑌) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
67 difsn 4802 . . . . . . . . . 10 𝐵𝑌 → (𝑌 ∖ {𝐵}) = 𝑌)
6866, 67eqtrid 2783 . . . . . . . . 9 𝐵𝑌 → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
6968ad2antlr 724 . . . . . . . 8 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
7055, 62, 693brtr3d 5180 . . . . . . 7 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑋𝑌)
7170expr 456 . . . . . 6 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑌 ∈ V) → (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋𝑌))
7271exlimdv 1935 . . . . 5 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑌 ∈ V) → (∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋𝑌))
737, 72syl5 34 . . . 4 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑌 ∈ V) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → 𝑋𝑌))
7473impancom 451 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → (𝑌 ∈ V → 𝑋𝑌))
756, 74mpd 15 . 2 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑋𝑌)
76 en2sn 9044 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴} ≈ {𝐵})
7732, 45, 76mp2an 689 . . . 4 {𝐴} ≈ {𝐵}
78 endom 8978 . . . 4 ({𝐴} ≈ {𝐵} → {𝐴} ≼ {𝐵})
7977, 78mp1i 13 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → {𝐴} ≼ {𝐵})
80 simpr 484 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → 𝑋𝑌)
81 incom 4202 . . . . 5 ({𝐵} ∩ 𝑌) = (𝑌 ∩ {𝐵})
82 disjsn 4716 . . . . . 6 ((𝑌 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝑌)
8382biimpri 227 . . . . 5 𝐵𝑌 → (𝑌 ∩ {𝐵}) = ∅)
8481, 83eqtrid 2783 . . . 4 𝐵𝑌 → ({𝐵} ∩ 𝑌) = ∅)
8584ad2antlr 724 . . 3 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → ({𝐵} ∩ 𝑌) = ∅)
86 undom 9062 . . 3 ((({𝐴} ≼ {𝐵} ∧ 𝑋𝑌) ∧ ({𝐵} ∩ 𝑌) = ∅) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
8779, 80, 85, 86syl21anc 835 . 2 (((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) ∧ 𝑋𝑌) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
8875, 87impbida 798 1 ((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1540  wex 1780  wcel 2105  Vcvv 3473  cdif 3946  cun 3947  cin 3948  wss 3949  c0 4323  {csn 4629   class class class wbr 5149  ccnv 5676  cres 5679  cima 5680  Fun wfun 6538   Fn wfn 6539  wf 6540  1-1wf1 6541  1-1-ontowf1o 6543  cfv 6544  cen 8939  cdom 8940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-en 8943  df-dom 8944
This theorem is referenced by:  domunfican  9323
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