Theorem enadj 6061

Description: Equivalence law for adjunction. Theorem XI.1.13 of [Rosser] p. 348. (Contributed by SF, 25-Feb-2015.)

Hypotheses

Ref	Expression
enadj.1	⊢ A ∈ V
enadj.2	⊢ B ∈ V
enadj.3	⊢ X ∈ V
enadj.4	⊢ Y ∈ V

Assertion

Ref	Expression
enadj	⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) → A ≈ B)

Proof of Theorem enadj

Step	Hyp	Ref	Expression
1		sneq 3745	. . . . . 6 ⊢ (X = Y → {X} = {Y})
2	1	uneq2d 3419	. . . . 5 ⊢ (X = Y → (A ∪ {X}) = (A ∪ {Y}))
3	2	eqeq1d 2361	. . . 4 ⊢ (X = Y → ((A ∪ {X}) = (B ∪ {Y}) ↔ (A ∪ {Y}) = (B ∪ {Y})))
4		eleq1 2413	. . . . 5 ⊢ (X = Y → (X ∈ A ↔ Y ∈ A))
5	4	notbid 285	. . . 4 ⊢ (X = Y → (¬ X ∈ A ↔ ¬ Y ∈ A))
6	3, 5	3anbi12d 1253	. . 3 ⊢ (X = Y → (((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ↔ ((A ∪ {Y}) = (B ∪ {Y}) ∧ ¬ Y ∈ A ∧ ¬ Y ∈ B)))
7		simp1 955	. . . . . 6 ⊢ (((A ∪ {Y}) = (B ∪ {Y}) ∧ ¬ Y ∈ A ∧ ¬ Y ∈ B) → (A ∪ {Y}) = (B ∪ {Y}))
8	7	difeq1d 3385	. . . . 5 ⊢ (((A ∪ {Y}) = (B ∪ {Y}) ∧ ¬ Y ∈ A ∧ ¬ Y ∈ B) → ((A ∪ {Y}) ∖ {Y}) = ((B ∪ {Y}) ∖ {Y}))
9		nnsucelrlem2 4426	. . . . . 6 ⊢ (¬ Y ∈ A → ((A ∪ {Y}) ∖ {Y}) = A)
10	9	3ad2ant2 977	. . . . 5 ⊢ (((A ∪ {Y}) = (B ∪ {Y}) ∧ ¬ Y ∈ A ∧ ¬ Y ∈ B) → ((A ∪ {Y}) ∖ {Y}) = A)
11		nnsucelrlem2 4426	. . . . . 6 ⊢ (¬ Y ∈ B → ((B ∪ {Y}) ∖ {Y}) = B)
12	11	3ad2ant3 978	. . . . 5 ⊢ (((A ∪ {Y}) = (B ∪ {Y}) ∧ ¬ Y ∈ A ∧ ¬ Y ∈ B) → ((B ∪ {Y}) ∖ {Y}) = B)
13	8, 10, 12	3eqtr3d 2393	. . . 4 ⊢ (((A ∪ {Y}) = (B ∪ {Y}) ∧ ¬ Y ∈ A ∧ ¬ Y ∈ B) → A = B)
14		enadj.2	. . . . 5 ⊢ B ∈ V
15	14	enrflx 6036	. . . 4 ⊢ B ≈ B
16	13, 15	syl6eqbr 4677	. . 3 ⊢ (((A ∪ {Y}) = (B ∪ {Y}) ∧ ¬ Y ∈ A ∧ ¬ Y ∈ B) → A ≈ B)
17	6, 16	syl6bi 219	. 2 ⊢ (X = Y → (((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) → A ≈ B))
18		elsni 3758	. . . . . . . . 9 ⊢ (Y ∈ {X} → Y = X)
19	18	eqcomd 2358	. . . . . . . 8 ⊢ (Y ∈ {X} → X = Y)
20	19	necon3ai 2557	. . . . . . 7 ⊢ (X ≠ Y → ¬ Y ∈ {X})
21	20	adantr 451	. . . . . 6 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → ¬ Y ∈ {X})
22		ssun2 3428	. . . . . . . . 9 ⊢ {Y} ⊆ (B ∪ {Y})
23		enadj.4	. . . . . . . . . 10 ⊢ Y ∈ V
24	23	snid 3761	. . . . . . . . 9 ⊢ Y ∈ {Y}
25	22, 24	sselii 3271	. . . . . . . 8 ⊢ Y ∈ (B ∪ {Y})
26		simpr1 961	. . . . . . . 8 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → (A ∪ {X}) = (B ∪ {Y}))
27	25, 26	syl5eleqr 2440	. . . . . . 7 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → Y ∈ (A ∪ {X}))
28		elun 3221	. . . . . . 7 ⊢ (Y ∈ (A ∪ {X}) ↔ (Y ∈ A ∨ Y ∈ {X}))
29	27, 28	sylib 188	. . . . . 6 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → (Y ∈ A ∨ Y ∈ {X}))
30		orel2 372	. . . . . 6 ⊢ (¬ Y ∈ {X} → ((Y ∈ A ∨ Y ∈ {X}) → Y ∈ A))
31	21, 29, 30	sylc 56	. . . . 5 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → Y ∈ A)
32		elsni 3758	. . . . . . . 8 ⊢ (X ∈ {Y} → X = Y)
33	32	necon3ai 2557	. . . . . . 7 ⊢ (X ≠ Y → ¬ X ∈ {Y})
34	33	adantr 451	. . . . . 6 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → ¬ X ∈ {Y})
35		ssun2 3428	. . . . . . . . 9 ⊢ {X} ⊆ (A ∪ {X})
36		enadj.3	. . . . . . . . . 10 ⊢ X ∈ V
37	36	snid 3761	. . . . . . . . 9 ⊢ X ∈ {X}
38	35, 37	sselii 3271	. . . . . . . 8 ⊢ X ∈ (A ∪ {X})
39	38, 26	syl5eleq 2439	. . . . . . 7 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → X ∈ (B ∪ {Y}))
40		elun 3221	. . . . . . 7 ⊢ (X ∈ (B ∪ {Y}) ↔ (X ∈ B ∨ X ∈ {Y}))
41	39, 40	sylib 188	. . . . . 6 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → (X ∈ B ∨ X ∈ {Y}))
42		orel2 372	. . . . . 6 ⊢ (¬ X ∈ {Y} → ((X ∈ B ∨ X ∈ {Y}) → X ∈ B))
43	34, 41, 42	sylc 56	. . . . 5 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → X ∈ B)
44	31, 43	jca 518	. . . 4 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → (Y ∈ A ∧ X ∈ B))
45		simpl1 958	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (A ∪ {X}) = (B ∪ {Y}))
46		simpl2 959	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ¬ X ∈ A)
47		simpl3 960	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ¬ Y ∈ B)
48		simprl 732	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → Y ∈ A)
49		simprr 733	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → X ∈ B)
50		enadjlem1 6060	. . . . . . 7 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (A ∖ {Y}) = (B ∖ {X}))
51	45, 46, 47, 48, 49, 50	syl122anc 1191	. . . . . 6 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (A ∖ {Y}) = (B ∖ {X}))
52	51	3adant1 973	. . . . 5 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (A ∖ {Y}) = (B ∖ {X}))
53		enadj.1	. . . . . . . . . . 11 ⊢ A ∈ V
54		snex 4112	. . . . . . . . . . 11 ⊢ {Y} ∈ V
55	53, 54	difex 4108	. . . . . . . . . 10 ⊢ (A ∖ {Y}) ∈ V
56	55	enrflx 6036	. . . . . . . . 9 ⊢ (A ∖ {Y}) ≈ (A ∖ {Y})
57		breq2 4644	. . . . . . . . 9 ⊢ ((A ∖ {Y}) = (B ∖ {X}) → ((A ∖ {Y}) ≈ (A ∖ {Y}) ↔ (A ∖ {Y}) ≈ (B ∖ {X})))
58	56, 57	mpbii 202	. . . . . . . 8 ⊢ ((A ∖ {Y}) = (B ∖ {X}) → (A ∖ {Y}) ≈ (B ∖ {X}))
59	58	adantl 452	. . . . . . 7 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → (A ∖ {Y}) ≈ (B ∖ {X}))
60	23, 36	ensn 6059	. . . . . . . 8 ⊢ {Y} ≈ {X}
61	60	a1i 10	. . . . . . 7 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → {Y} ≈ {X})
62		incom 3449	. . . . . . . . 9 ⊢ ((A ∖ {Y}) ∩ {Y}) = ({Y} ∩ (A ∖ {Y}))
63		disjdif 3623	. . . . . . . . 9 ⊢ ({Y} ∩ (A ∖ {Y})) = ∅
64	62, 63	eqtri 2373	. . . . . . . 8 ⊢ ((A ∖ {Y}) ∩ {Y}) = ∅
65	64	a1i 10	. . . . . . 7 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → ((A ∖ {Y}) ∩ {Y}) = ∅)
66		incom 3449	. . . . . . . . 9 ⊢ ((B ∖ {X}) ∩ {X}) = ({X} ∩ (B ∖ {X}))
67		disjdif 3623	. . . . . . . . 9 ⊢ ({X} ∩ (B ∖ {X})) = ∅
68	66, 67	eqtri 2373	. . . . . . . 8 ⊢ ((B ∖ {X}) ∩ {X}) = ∅
69	68	a1i 10	. . . . . . 7 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → ((B ∖ {X}) ∩ {X}) = ∅)
70		unen 6049	. . . . . . 7 ⊢ ((((A ∖ {Y}) ≈ (B ∖ {X}) ∧ {Y} ≈ {X}) ∧ (((A ∖ {Y}) ∩ {Y}) = ∅ ∧ ((B ∖ {X}) ∩ {X}) = ∅)) → ((A ∖ {Y}) ∪ {Y}) ≈ ((B ∖ {X}) ∪ {X}))
71	59, 61, 65, 69, 70	syl22anc 1183	. . . . . 6 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → ((A ∖ {Y}) ∪ {Y}) ≈ ((B ∖ {X}) ∪ {X}))
72		simpl3l 1010	. . . . . . 7 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → Y ∈ A)
73		nnsucelrlem4 4428	. . . . . . 7 ⊢ (Y ∈ A → ((A ∖ {Y}) ∪ {Y}) = A)
74	72, 73	syl 15	. . . . . 6 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → ((A ∖ {Y}) ∪ {Y}) = A)
75		simpl3r 1011	. . . . . . 7 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → X ∈ B)
76		nnsucelrlem4 4428	. . . . . . 7 ⊢ (X ∈ B → ((B ∖ {X}) ∪ {X}) = B)
77	75, 76	syl 15	. . . . . 6 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → ((B ∖ {X}) ∪ {X}) = B)
78	71, 74, 77	3brtr3d 4669	. . . . 5 ⊢ (((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (A ∖ {Y}) = (B ∖ {X})) → A ≈ B)
79	52, 78	mpdan 649	. . . 4 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → A ≈ B)
80	44, 79	mpd3an3 1278	. . 3 ⊢ ((X ≠ Y ∧ ((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B)) → A ≈ B)
81	80	ex 423	. 2 ⊢ (X ≠ Y → (((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) → A ≈ B))
82	17, 81	pm2.61ine 2593	1 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ ¬ X ∈ A ∧ ¬ Y ∈ B) → A ≈ B)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738   class class class wbr 4640   ≈ cen 6029

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-en 6030

This theorem is used by:  peano4nc  6151