Theorem ncdisjun 6137

Description: Cardinality of disjoint union of two sets. (Contributed by SF, 24-Feb-2015.)

Hypotheses

Ref	Expression
ncdisjun.1	⊢ A ∈ V
ncdisjun.2	⊢ B ∈ V

Assertion

Ref	Expression
ncdisjun	⊢ ((A ∩ B) = ∅ → Nc (A ∪ B) = ( Nc A +c Nc B))

Proof of Theorem ncdisjun

Dummy variables p q r x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnc 6126	. . 3 ⊢ (x ∈ Nc (A ∪ B) ↔ x ≈ (A ∪ B))
2		bren 6031	. . . . 5 ⊢ (x ≈ (A ∪ B) ↔ ∃r r:x–1-1-onto→(A ∪ B))
3		f1ocnv 5300	. . . . . . 7 ⊢ (r:x–1-1-onto→(A ∪ B) → ◡r:(A ∪ B)–1-1-onto→x)
4		imaundi 5040	. . . . . . . . . . 11 ⊢ (◡r “ (A ∪ B)) = ((◡r “ A) ∪ (◡r “ B))
5		imadmrn 5009	. . . . . . . . . . . . 13 ⊢ (◡r “ dom ◡r) = ran ◡r
6	5	a1i 10	. . . . . . . . . . . 12 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r “ dom ◡r) = ran ◡r)
7		f1odm 5291	. . . . . . . . . . . . 13 ⊢ (◡r:(A ∪ B)–1-1-onto→x → dom ◡r = (A ∪ B))
8	7	imaeq2d 4943	. . . . . . . . . . . 12 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r “ dom ◡r) = (◡r “ (A ∪ B)))
9		f1ofo 5294	. . . . . . . . . . . . 13 ⊢ (◡r:(A ∪ B)–1-1-onto→x → ◡r:(A ∪ B)–onto→x)
10		forn 5273	. . . . . . . . . . . . 13 ⊢ (◡r:(A ∪ B)–onto→x → ran ◡r = x)
11	9, 10	syl 15	. . . . . . . . . . . 12 ⊢ (◡r:(A ∪ B)–1-1-onto→x → ran ◡r = x)
12	6, 8, 11	3eqtr3d 2393	. . . . . . . . . . 11 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r “ (A ∪ B)) = x)
13	4, 12	syl5eqr 2399	. . . . . . . . . 10 ⊢ (◡r:(A ∪ B)–1-1-onto→x → ((◡r “ A) ∪ (◡r “ B)) = x)
14	13	adantl 452	. . . . . . . . 9 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → ((◡r “ A) ∪ (◡r “ B)) = x)
15		f1of1 5287	. . . . . . . . . . . . . 14 ⊢ (◡r:(A ∪ B)–1-1-onto→x → ◡r:(A ∪ B)–1-1→x)
16		ssun1 3427	. . . . . . . . . . . . . 14 ⊢ A ⊆ (A ∪ B)
17		f1ores 5301	. . . . . . . . . . . . . 14 ⊢ ((◡r:(A ∪ B)–1-1→x ∧ A ⊆ (A ∪ B)) → (◡r ↾ A):A–1-1-onto→(◡r “ A))
18	15, 16, 17	sylancl 643	. . . . . . . . . . . . 13 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r ↾ A):A–1-1-onto→(◡r “ A))
19		f1ocnv 5300	. . . . . . . . . . . . 13 ⊢ ((◡r ↾ A):A–1-1-onto→(◡r “ A) → ◡(◡r ↾ A):(◡r “ A)–1-1-onto→A)
20		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ r ∈ V
21	20	cnvex 5103	. . . . . . . . . . . . . . . 16 ⊢ ◡r ∈ V
22		ncdisjun.1	. . . . . . . . . . . . . . . 16 ⊢ A ∈ V
23	21, 22	resex 5118	. . . . . . . . . . . . . . 15 ⊢ (◡r ↾ A) ∈ V
24	23	cnvex 5103	. . . . . . . . . . . . . 14 ⊢ ◡(◡r ↾ A) ∈ V
25	24	f1oen 6034	. . . . . . . . . . . . 13 ⊢ (◡(◡r ↾ A):(◡r “ A)–1-1-onto→A → (◡r “ A) ≈ A)
26	18, 19, 25	3syl 18	. . . . . . . . . . . 12 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r “ A) ≈ A)
27		elnc 6126	. . . . . . . . . . . 12 ⊢ ((◡r “ A) ∈ Nc A ↔ (◡r “ A) ≈ A)
28	26, 27	sylibr 203	. . . . . . . . . . 11 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r “ A) ∈ Nc A)
29	28	adantl 452	. . . . . . . . . 10 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → (◡r “ A) ∈ Nc A)
30		ssun2 3428	. . . . . . . . . . . . . 14 ⊢ B ⊆ (A ∪ B)
31		f1ores 5301	. . . . . . . . . . . . . 14 ⊢ ((◡r:(A ∪ B)–1-1→x ∧ B ⊆ (A ∪ B)) → (◡r ↾ B):B–1-1-onto→(◡r “ B))
32	15, 30, 31	sylancl 643	. . . . . . . . . . . . 13 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r ↾ B):B–1-1-onto→(◡r “ B))
33		f1ocnv 5300	. . . . . . . . . . . . 13 ⊢ ((◡r ↾ B):B–1-1-onto→(◡r “ B) → ◡(◡r ↾ B):(◡r “ B)–1-1-onto→B)
34		ncdisjun.2	. . . . . . . . . . . . . . . 16 ⊢ B ∈ V
35	21, 34	resex 5118	. . . . . . . . . . . . . . 15 ⊢ (◡r ↾ B) ∈ V
36	35	cnvex 5103	. . . . . . . . . . . . . 14 ⊢ ◡(◡r ↾ B) ∈ V
37	36	f1oen 6034	. . . . . . . . . . . . 13 ⊢ (◡(◡r ↾ B):(◡r “ B)–1-1-onto→B → (◡r “ B) ≈ B)
38	32, 33, 37	3syl 18	. . . . . . . . . . . 12 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r “ B) ≈ B)
39	38	adantl 452	. . . . . . . . . . 11 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → (◡r “ B) ≈ B)
40		elnc 6126	. . . . . . . . . . 11 ⊢ ((◡r “ B) ∈ Nc B ↔ (◡r “ B) ≈ B)
41	39, 40	sylibr 203	. . . . . . . . . 10 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → (◡r “ B) ∈ Nc B)
42		df-f1 4793	. . . . . . . . . . . . . 14 ⊢ (◡r:(A ∪ B)–1-1→x ↔ (◡r:(A ∪ B)–→x ∧ Fun ◡◡r))
43	42	simprbi 450	. . . . . . . . . . . . 13 ⊢ (◡r:(A ∪ B)–1-1→x → Fun ◡◡r)
44		imain 5173	. . . . . . . . . . . . 13 ⊢ (Fun ◡◡r → (◡r “ (A ∩ B)) = ((◡r “ A) ∩ (◡r “ B)))
45	15, 43, 44	3syl 18	. . . . . . . . . . . 12 ⊢ (◡r:(A ∪ B)–1-1-onto→x → (◡r “ (A ∩ B)) = ((◡r “ A) ∩ (◡r “ B)))
46	45	adantl 452	. . . . . . . . . . 11 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → (◡r “ (A ∩ B)) = ((◡r “ A) ∩ (◡r “ B)))
47		imaeq2 4939	. . . . . . . . . . . . 13 ⊢ ((A ∩ B) = ∅ → (◡r “ (A ∩ B)) = (◡r “ ∅))
48		ima0 5014	. . . . . . . . . . . . 13 ⊢ (◡r “ ∅) = ∅
49	47, 48	syl6eq 2401	. . . . . . . . . . . 12 ⊢ ((A ∩ B) = ∅ → (◡r “ (A ∩ B)) = ∅)
50	49	adantr 451	. . . . . . . . . . 11 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → (◡r “ (A ∩ B)) = ∅)
51	46, 50	eqtr3d 2387	. . . . . . . . . 10 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → ((◡r “ A) ∩ (◡r “ B)) = ∅)
52		eladdci 4400	. . . . . . . . . 10 ⊢ (((◡r “ A) ∈ Nc A ∧ (◡r “ B) ∈ Nc B ∧ ((◡r “ A) ∩ (◡r “ B)) = ∅) → ((◡r “ A) ∪ (◡r “ B)) ∈ ( Nc A +c Nc B))
53	29, 41, 51, 52	syl3anc 1182	. . . . . . . . 9 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → ((◡r “ A) ∪ (◡r “ B)) ∈ ( Nc A +c Nc B))
54	14, 53	eqeltrrd 2428	. . . . . . . 8 ⊢ (((A ∩ B) = ∅ ∧ ◡r:(A ∪ B)–1-1-onto→x) → x ∈ ( Nc A +c Nc B))
55	54	ex 423	. . . . . . 7 ⊢ ((A ∩ B) = ∅ → (◡r:(A ∪ B)–1-1-onto→x → x ∈ ( Nc A +c Nc B)))
56	3, 55	syl5 28	. . . . . 6 ⊢ ((A ∩ B) = ∅ → (r:x–1-1-onto→(A ∪ B) → x ∈ ( Nc A +c Nc B)))
57	56	exlimdv 1636	. . . . 5 ⊢ ((A ∩ B) = ∅ → (∃r r:x–1-1-onto→(A ∪ B) → x ∈ ( Nc A +c Nc B)))
58	2, 57	syl5bi 208	. . . 4 ⊢ ((A ∩ B) = ∅ → (x ≈ (A ∪ B) → x ∈ ( Nc A +c Nc B)))
59		eladdc 4399	. . . . 5 ⊢ (x ∈ ( Nc A +c Nc B) ↔ ∃p ∈ Nc A∃q ∈ Nc B((p ∩ q) = ∅ ∧ x = (p ∪ q)))
60		simplrl 736	. . . . . . . . . 10 ⊢ ((((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) ∧ (p ∩ q) = ∅) → p ∈ Nc A)
61		elnc 6126	. . . . . . . . . 10 ⊢ (p ∈ Nc A ↔ p ≈ A)
62	60, 61	sylib 188	. . . . . . . . 9 ⊢ ((((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) ∧ (p ∩ q) = ∅) → p ≈ A)
63		simplrr 737	. . . . . . . . . 10 ⊢ ((((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) ∧ (p ∩ q) = ∅) → q ∈ Nc B)
64		elnc 6126	. . . . . . . . . 10 ⊢ (q ∈ Nc B ↔ q ≈ B)
65	63, 64	sylib 188	. . . . . . . . 9 ⊢ ((((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) ∧ (p ∩ q) = ∅) → q ≈ B)
66		simpr 447	. . . . . . . . 9 ⊢ ((((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) ∧ (p ∩ q) = ∅) → (p ∩ q) = ∅)
67		simpll 730	. . . . . . . . 9 ⊢ ((((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) ∧ (p ∩ q) = ∅) → (A ∩ B) = ∅)
68		unen 6049	. . . . . . . . 9 ⊢ (((p ≈ A ∧ q ≈ B) ∧ ((p ∩ q) = ∅ ∧ (A ∩ B) = ∅)) → (p ∪ q) ≈ (A ∪ B))
69	62, 65, 66, 67, 68	syl22anc 1183	. . . . . . . 8 ⊢ ((((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) ∧ (p ∩ q) = ∅) → (p ∪ q) ≈ (A ∪ B))
70		breq1 4643	. . . . . . . 8 ⊢ (x = (p ∪ q) → (x ≈ (A ∪ B) ↔ (p ∪ q) ≈ (A ∪ B)))
71	69, 70	syl5ibrcom 213	. . . . . . 7 ⊢ ((((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) ∧ (p ∩ q) = ∅) → (x = (p ∪ q) → x ≈ (A ∪ B)))
72	71	expimpd 586	. . . . . 6 ⊢ (((A ∩ B) = ∅ ∧ (p ∈ Nc A ∧ q ∈ Nc B)) → (((p ∩ q) = ∅ ∧ x = (p ∪ q)) → x ≈ (A ∪ B)))
73	72	rexlimdvva 2746	. . . . 5 ⊢ ((A ∩ B) = ∅ → (∃p ∈ Nc A∃q ∈ Nc B((p ∩ q) = ∅ ∧ x = (p ∪ q)) → x ≈ (A ∪ B)))
74	59, 73	syl5bi 208	. . . 4 ⊢ ((A ∩ B) = ∅ → (x ∈ ( Nc A +c Nc B) → x ≈ (A ∪ B)))
75	58, 74	impbid 183	. . 3 ⊢ ((A ∩ B) = ∅ → (x ≈ (A ∪ B) ↔ x ∈ ( Nc A +c Nc B)))
76	1, 75	syl5bb 248	. 2 ⊢ ((A ∩ B) = ∅ → (x ∈ Nc (A ∪ B) ↔ x ∈ ( Nc A +c Nc B)))
77	76	eqrdv 2351	1 ⊢ ((A ∩ B) = ∅ → Nc (A ∪ B) = ( Nc A +c Nc B))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551   +_c cplc 4376   class class class wbr 4640   “ cima 4723  ^◡ccnv 4772  dom cdm 4773  ran crn 4774   ↾ cres 4775  Fun wfun 4776  –→wf 4778  –1-1→wf1 4779  –onto→wfo 4780  –1-1-onto→wf1o 4781   ≈ cen 6029   Nc cnc 6092

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-si 4729  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-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-en 6030  df-nc 6102

This theorem is used by:  ncaddccl  6145  1p1e2c  6156  2p1e3c  6157  dflec2  6211  addcdi  6251  nchoicelem7  6296  nchoicelem14  6303