Theorem domunsncan 9091

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 4159	. . . 4 ⊢ 𝑌 ⊆ ({𝐵} ∪ 𝑌)
2		reldom 8970	. . . . . 6 ⊢ Rel ≼
3	2	brrelex2i 5716	. . . . 5 ⊢ (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ({𝐵} ∪ 𝑌) ∈ V)
4	3	adantl 481	. . . 4 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → ({𝐵} ∪ 𝑌) ∈ V)
5		ssexg 5298	. . . 4 ⊢ ((𝑌 ⊆ ({𝐵} ∪ 𝑌) ∧ ({𝐵} ∪ 𝑌) ∈ V) → 𝑌 ∈ V)
6	1, 4, 5	sylancr 587	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑌 ∈ V)
7		brdomi 8978	. . . . 5 ⊢ (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → ∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
8		vex 3468	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
9	8	resex 6021	. . . . . . . . . 10 ⊢ (𝑓 ↾ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∈ V
10		simprr 772	. . . . . . . . . . 11 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))
11		difss 4116	. . . . . . . . . . 11 ⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) ⊆ ({𝐴} ∪ 𝑋)
12		f1ores 6837	. . . . . . . . . . 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 586	. . . . . . . . . 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 8986	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})))
16		df-f1 6541	. . . . . . . . . . . 12 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) ↔ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ Fun ◡𝑓))
17		imadif 6625	. . . . . . . . . . . 12 ⊢ (Fun ◡𝑓 → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
18	16, 17	simplbiim 504	. . . . . . . . . . 11 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) = ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})))
19	18	ad2antll 729	. . . . . . . . . 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 5411	. . . . . . . . . . . . . 14 ⊢ {𝐵} ∈ V
21		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑌 ∈ V)
22		unexg 7742	. . . . . . . . . . . . . 14 ⊢ (({𝐵} ∈ V ∧ 𝑌 ∈ V) → ({𝐵} ∪ 𝑌) ∈ V)
23	20, 21, 22	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ({𝐵} ∪ 𝑌) ∈ V)
24	23	difexd 5306	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∈ V)
25		f1f 6779	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌))
26		fimass 6731	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
27	25, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
28	27	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ ({𝐴} ∪ 𝑋)) ⊆ ({𝐵} ∪ 𝑌))
29	28	ssdifd 4125	. . . . . . . . . . . . 13 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
30		f1fn 6780	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑓 Fn ({𝐴} ∪ 𝑋))
31	30	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑓 Fn ({𝐴} ∪ 𝑋))
32		domunsncan.a	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 ∈ V
33	32	snid 4643	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 ∈ {𝐴}
34		elun1 4162	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝑋))
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝑋)
36		fnsnfv 6963	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn ({𝐴} ∪ 𝑋) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
37	31, 35, 36	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
38	37	difeq2d 4106	. . . . . . . . . . . . 13 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) = (({𝐵} ∪ 𝑌) ∖ (𝑓 “ {𝐴})))
39	29, 38	sseqtrrd 4001	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}))
40		ssdomg 9019	. . . . . . . . . . . 12 ⊢ ((({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ∈ V → (((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ⊆ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)})))
41	24, 39, 40	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)}))
42		ffvelcdm 7076	. . . . . . . . . . . . . 14 ⊢ ((𝑓:({𝐴} ∪ 𝑋)⟶({𝐵} ∪ 𝑌) ∧ 𝐴 ∈ ({𝐴} ∪ 𝑋)) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌))
43	25, 35, 42	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌))
44	43	ad2antll 729	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌))
45		domunsncan.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
46	45	snid 4643	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ {𝐵}
47		elun1 4162	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ ({𝐵} ∪ 𝑌))
48	46, 47	mp1i 13	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝐵 ∈ ({𝐵} ∪ 𝑌))
49		difsnen 9072	. . . . . . . . . . . 12 ⊢ ((({𝐵} ∪ 𝑌) ∈ V ∧ (𝑓‘𝐴) ∈ ({𝐵} ∪ 𝑌) ∧ 𝐵 ∈ ({𝐵} ∪ 𝑌)) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
50	23, 44, 48, 49	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {(𝑓‘𝐴)}) ≈ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
51		domentr 9032	. . . . . . . . . . 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}))
52	41, 50, 51	syl2anc 584	. . . . . . . . . 10 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → ((𝑓 “ ({𝐴} ∪ 𝑋)) ∖ (𝑓 “ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
53	19, 52	eqbrtrd 5146	. . . . . . . . 9 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
54		endomtr 9031	. . . . . . . . 9 ⊢ (((({𝐴} ∪ 𝑋) ∖ {𝐴}) ≈ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ∧ (𝑓 “ (({𝐴} ∪ 𝑋) ∖ {𝐴})) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵})) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
55	15, 53, 54	syl2anc 584	. . . . . . . 8 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) ≼ (({𝐵} ∪ 𝑌) ∖ {𝐵}))
56		uncom 4138	. . . . . . . . . . . 12 ⊢ ({𝐴} ∪ 𝑋) = (𝑋 ∪ {𝐴})
57	56	difeq1i 4102	. . . . . . . . . . 11 ⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) = ((𝑋 ∪ {𝐴}) ∖ {𝐴})
58		difun2 4461	. . . . . . . . . . 11 ⊢ ((𝑋 ∪ {𝐴}) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
59	57, 58	eqtri 2759	. . . . . . . . . 10 ⊢ (({𝐴} ∪ 𝑋) ∖ {𝐴}) = (𝑋 ∖ {𝐴})
60		difsn 4779	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ 𝑋 → (𝑋 ∖ {𝐴}) = 𝑋)
61	59, 60	eqtrid 2783	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ 𝑋 → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
62	61	ad2antrr 726	. . . . . . . 8 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐴} ∪ 𝑋) ∖ {𝐴}) = 𝑋)
63		uncom 4138	. . . . . . . . . . . 12 ⊢ ({𝐵} ∪ 𝑌) = (𝑌 ∪ {𝐵})
64	63	difeq1i 4102	. . . . . . . . . . 11 ⊢ (({𝐵} ∪ 𝑌) ∖ {𝐵}) = ((𝑌 ∪ {𝐵}) ∖ {𝐵})
65		difun2 4461	. . . . . . . . . . 11 ⊢ ((𝑌 ∪ {𝐵}) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
66	64, 65	eqtri 2759	. . . . . . . . . 10 ⊢ (({𝐵} ∪ 𝑌) ∖ {𝐵}) = (𝑌 ∖ {𝐵})
67		difsn 4779	. . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ 𝑌 → (𝑌 ∖ {𝐵}) = 𝑌)
68	66, 67	eqtrid 2783	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ 𝑌 → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
69	68	ad2antlr 727	. . . . . . . 8 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → (({𝐵} ∪ 𝑌) ∖ {𝐵}) = 𝑌)
70	55, 62, 69	3brtr3d 5155	. . . . . . 7 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ (𝑌 ∈ V ∧ 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌))) → 𝑋 ≼ 𝑌)
71	70	expr 456	. . . . . 6 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌))
72	71	exlimdv 1933	. . . . 5 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (∃𝑓 𝑓:({𝐴} ∪ 𝑋)–1-1→({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌))
73	7, 72	syl5 34	. . . 4 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑌 ∈ V) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) → 𝑋 ≼ 𝑌))
74	73	impancom 451	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → (𝑌 ∈ V → 𝑋 ≼ 𝑌))
75	6, 74	mpd 15	. 2 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌)) → 𝑋 ≼ 𝑌)
76		en2sn 9060	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴} ≈ {𝐵})
77	32, 45, 76	mp2an 692	. . . 4 ⊢ {𝐴} ≈ {𝐵}
78		endom 8998	. . . 4 ⊢ ({𝐴} ≈ {𝐵} → {𝐴} ≼ {𝐵})
79	77, 78	mp1i 13	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → {𝐴} ≼ {𝐵})
80		simpr 484	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → 𝑋 ≼ 𝑌)
81		incom 4189	. . . . 5 ⊢ ({𝐵} ∩ 𝑌) = (𝑌 ∩ {𝐵})
82		disjsn 4692	. . . . . 6 ⊢ ((𝑌 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝑌)
83	82	biimpri 228	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝑌 → (𝑌 ∩ {𝐵}) = ∅)
84	81, 83	eqtrid 2783	. . . 4 ⊢ (¬ 𝐵 ∈ 𝑌 → ({𝐵} ∩ 𝑌) = ∅)
85	84	ad2antlr 727	. . 3 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → ({𝐵} ∩ 𝑌) = ∅)
86		undom 9078	. . 3 ⊢ ((({𝐴} ≼ {𝐵} ∧ 𝑋 ≼ 𝑌) ∧ ({𝐵} ∩ 𝑌) = ∅) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
87	79, 80, 85, 86	syl21anc 837	. 2 ⊢ (((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) ∧ 𝑋 ≼ 𝑌) → ({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌))
88	75, 87	impbida 800	1 ⊢ ((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋 ≼ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3464   ∖ cdif 3928   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606   class class class wbr 5124  ^◡ccnv 5658   ↾ cres 5661   “ cima 5662  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536   ≈ cen 8961   ≼ cdom 8962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-en 8965  df-dom 8966

This theorem is referenced by:  domunfican  9338