Theorem domunsncan 8060

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	⊢ ((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋 ≼ 𝑌))

Proof of Theorem domunsncan

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun2 3777	. . . 4 ⊢ 𝑌 ⊆ ({𝐵} ∪ 𝑌)
2		reldom 7961	. . . . . 6 ⊢ Rel ≼
3	2	brrelex2i 5159	. . . . 5 ⊢ (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ({𝐵} ∪ 𝑌) ∈ V)
4	3	adantl 482	. . . 4 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → ({𝐵} ∪ 𝑌) ∈ V)
5		ssexg 4804	. . . 4 ⊢ ((𝑌 ⊆ ({𝐵} ∪ 𝑌) ∧ ({𝐵} ∪ 𝑌) ∈ V) → 𝑌 ∈ V)
6	1, 4, 5	sylancr 695	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑌 ∈ V)
7		brdomi 7966	. . . . 5 ⊢ (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
8		vex 3203	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
9	8	resex 5443	. . . . . . . . . 10 ⊢ (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∈ V
10		simprr 796	. . . . . . . . . . 11 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
11		difss 3737	. . . . . . . . . . 11 ⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) ⊆ ({𝐴} ∪ 𝑋)
12		f1ores 6151	. . . . . . . . . . 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 694	. . . . . . . . . 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 7971	. . . . . . . . . 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 695	. . . . . . . . 9 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})))
16		df-f1 5893	. . . . . . . . . . . . 13 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) ↔ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ Fun ◡𝑓))
17	16	simprbi 480	. . . . . . . . . . . 12 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → Fun ◡𝑓)
18		imadif 5973	. . . . . . . . . . . 12 ⊢ (Fun ◡𝑓 → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
20	19	ad2antll 765	. . . . . . . . . 10 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
21		snex 4908	. . . . . . . . . . . . . 14 ⊢ {𝐵} ∈ V
22		simprl 794	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑌 ∈ V)
23		unexg 6959	. . . . . . . . . . . . . 14 ⊢ (({𝐵} ∈ V ∧ 𝑌 ∈ V) → ({𝐵} ∪ 𝑌) ∈ V)
24	21, 22, 23	sylancr 695	. . . . . . . . . . . . 13 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ({𝐵} ∪ 𝑌) ∈ V)
25		difexg 4808	. . . . . . . . . . . . 13 ⊢ (({𝐵} ∪ 𝑌) ∈ V → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∈ V)
26	24, 25	syl 17	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∈ V)
27		f1f 6101	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌))
28		imassrn 5477	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ran 𝑓
29		frn 6053	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) → ran 𝑓 ⊆ ({𝐵} ∪ 𝑌))
30	28, 29	syl5ss 3614	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
31	27, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
32	31	ad2antll 765	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
33	32	ssdifd 3746	. . . . . . . . . . . . 13 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
34		f1fn 6102	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓 Fn ({𝐴} ∪ 𝑋))
35	34	ad2antll 765	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓 Fn ({𝐴} ∪ 𝑋))
36		domunsncan.a	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 ∈ V
37	36	snid 4208	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 ∈ {𝐴}
38		elun1 3780	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝑋))
39	37, 38	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝑋)
40		fnsnfv 6258	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn ({𝐴} ∪ 𝑋) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
41	35, 39, 40	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
42	41	difeq2d 3728	. . . . . . . . . . . . 13 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) = (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
43	33, 42	sseqtr4d 3642	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}))
44		ssdomg 8001	. . . . . . . . . . . 12 ⊢ ((({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∈ V → (((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)})))
45	26, 43, 44	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)}))
46		ffvelrn 6357	. . . . . . . . . . . . . 14 ⊢ ((𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌))
47	27, 39, 46	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌))
48	47	ad2antll 765	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌))
49		domunsncan.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
50	49	snid 4208	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ {𝐵}
51		elun1 3780	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ ({𝐵} ∪ 𝑌))
52	50, 51	mp1i 13	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝐵 ∈ ({𝐵} ∪ 𝑌))
53		difsnen 8042	. . . . . . . . . . . 12 ⊢ ((({𝐵} ∪ 𝑌) ∈ V ∧ (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌) ∧ 𝐵 ∈ ({𝐵} ∪ 𝑌)) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
54	24, 48, 52, 53	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
55		domentr 8015	. . . . . . . . . . 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}))
56	45, 54, 55	syl2anc 693	. . . . . . . . . 10 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
57	20, 56	eqbrtrd 4675	. . . . . . . . 9 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
58		endomtr 8014	. . . . . . . . 9 ⊢ (((({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∧ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
59	15, 57, 58	syl2anc 693	. . . . . . . 8 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
60		uncom 3757	. . . . . . . . . . . 12 ⊢ ({𝐴} ∪ 𝑋) = (𝑋 ∪ {𝐴})
61	60	difeq1i 3724	. . . . . . . . . . 11 ⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) = ((𝑋 ∪ {𝐴}) ∖ {𝐴})
62		difun2 4048	. . . . . . . . . . 11 ⊢ ((𝑋 ∪ {𝐴}) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
63	61, 62	eqtri 2644	. . . . . . . . . 10 ⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
64		difsn 4328	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ 𝑋 → (𝑋 ∖ {𝐴}) = 𝑋)
65	63, 64	syl5eq 2668	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ 𝑋 → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
66	65	ad2antrr 762	. . . . . . . 8 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
67		uncom 3757	. . . . . . . . . . . 12 ⊢ ({𝐵} ∪ 𝑌) = (𝑌 ∪ {𝐵})
68	67	difeq1i 3724	. . . . . . . . . . 11 ⊢ (({𝐵} ∪ 𝑌) ∖ {𝐵}) = ((𝑌 ∪ {𝐵}) ∖ {𝐵})
69		difun2 4048	. . . . . . . . . . 11 ⊢ ((𝑌 ∪ {𝐵}) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
70	68, 69	eqtri 2644	. . . . . . . . . 10 ⊢ (({𝐵} ∪ 𝑌) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
71		difsn 4328	. . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ 𝑌 → (𝑌 ∖ {𝐵}) = 𝑌)
72	70, 71	syl5eq 2668	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ 𝑌 → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
73	72	ad2antlr 763	. . . . . . . 8 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
74	59, 66, 73	3brtr3d 4684	. . . . . . 7 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑋 ≼ 𝑌)
75	74	expr 643	. . . . . 6 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌))
76	75	exlimdv 1861	. . . . 5 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌))
77	7, 76	syl5 34	. . . 4 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌))
78	77	impancom 456	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → (𝑌 ∈ V → 𝑋 ≼ 𝑌))
79	6, 78	mpd 15	. 2 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑋 ≼ 𝑌)
80		en2sn 8037	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴} ≈ {𝐵})
81	36, 49, 80	mp2an 708	. . . 4 ⊢ {𝐴} ≈ {𝐵}
82		endom 7982	. . . 4 ⊢ ({𝐴} ≈ {𝐵} → {𝐴} ≼ {𝐵})
83	81, 82	mp1i 13	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → {𝐴} ≼ {𝐵})
84		simpr 477	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → 𝑋 ≼ 𝑌)
85		incom 3805	. . . . 5 ⊢ ({𝐵} ∩ 𝑌) = (𝑌 ∩ {𝐵})
86		disjsn 4246	. . . . . 6 ⊢ ((𝑌 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝑌)
87	86	biimpri 218	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝑌 → (𝑌 ∩ {𝐵}) = ∅)
88	85, 87	syl5eq 2668	. . . 4 ⊢ (¬ 𝐵 ∈ 𝑌 → ({𝐵} ∩ 𝑌) = ∅)
89	88	ad2antlr 763	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → ({𝐵} ∩ 𝑌) = ∅)
90		undom 8048	. . 3 ⊢ ((({𝐴} ≼ {𝐵} ∧ 𝑋 ≼ 𝑌) ∧ ({𝐵} ∩ 𝑌) = ∅) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
91	83, 84, 89, 90	syl21anc 1325	. 2 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
92	79, 91	impbida 877	1 ⊢ ((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋 ≼ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888   ≈ cen 7952   ≼ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957

This theorem is referenced by:  domunfican  8233