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Theorem ncdisjun 6136
 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 ((AB) = Nc (AB) = ( Nc A +c Nc B))

Proof of Theorem ncdisjun
Dummy variables p q r x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnc 6125 . . 3 (x Nc (AB) ↔ x ≈ (AB))
2 bren 6030 . . . . 5 (x ≈ (AB) ↔ r r:x1-1-onto→(AB))
3 f1ocnv 5299 . . . . . . 7 (r:x1-1-onto→(AB) → r:(AB)–1-1-ontox)
4 imaundi 5039 . . . . . . . . . . 11 (r “ (AB)) = ((rA) ∪ (rB))
5 imadmrn 5008 . . . . . . . . . . . . 13 (r “ dom r) = ran r
65a1i 10 . . . . . . . . . . . 12 (r:(AB)–1-1-ontox → (r “ dom r) = ran r)
7 f1odm 5290 . . . . . . . . . . . . 13 (r:(AB)–1-1-ontox → dom r = (AB))
87imaeq2d 4942 . . . . . . . . . . . 12 (r:(AB)–1-1-ontox → (r “ dom r) = (r “ (AB)))
9 f1ofo 5293 . . . . . . . . . . . . 13 (r:(AB)–1-1-ontoxr:(AB)–ontox)
10 forn 5272 . . . . . . . . . . . . 13 (r:(AB)–ontox → ran r = x)
119, 10syl 15 . . . . . . . . . . . 12 (r:(AB)–1-1-ontox → ran r = x)
126, 8, 113eqtr3d 2393 . . . . . . . . . . 11 (r:(AB)–1-1-ontox → (r “ (AB)) = x)
134, 12syl5eqr 2399 . . . . . . . . . 10 (r:(AB)–1-1-ontox → ((rA) ∪ (rB)) = x)
1413adantl 452 . . . . . . . . 9 (((AB) = r:(AB)–1-1-ontox) → ((rA) ∪ (rB)) = x)
15 f1of1 5286 . . . . . . . . . . . . . 14 (r:(AB)–1-1-ontoxr:(AB)–1-1x)
16 ssun1 3426 . . . . . . . . . . . . . 14 A (AB)
17 f1ores 5300 . . . . . . . . . . . . . 14 ((r:(AB)–1-1x A (AB)) → (r A):A1-1-onto→(rA))
1815, 16, 17sylancl 643 . . . . . . . . . . . . 13 (r:(AB)–1-1-ontox → (r A):A1-1-onto→(rA))
19 f1ocnv 5299 . . . . . . . . . . . . 13 ((r A):A1-1-onto→(rA) → (r A):(rA)–1-1-ontoA)
20 vex 2862 . . . . . . . . . . . . . . . . 17 r V
2120cnvex 5102 . . . . . . . . . . . . . . . 16 r V
22 ncdisjun.1 . . . . . . . . . . . . . . . 16 A V
2321, 22resex 5117 . . . . . . . . . . . . . . 15 (r A) V
2423cnvex 5102 . . . . . . . . . . . . . 14 (r A) V
2524f1oen 6033 . . . . . . . . . . . . 13 ((r A):(rA)–1-1-ontoA → (rA) ≈ A)
2618, 19, 253syl 18 . . . . . . . . . . . 12 (r:(AB)–1-1-ontox → (rA) ≈ A)
27 elnc 6125 . . . . . . . . . . . 12 ((rA) Nc A ↔ (rA) ≈ A)
2826, 27sylibr 203 . . . . . . . . . . 11 (r:(AB)–1-1-ontox → (rA) Nc A)
2928adantl 452 . . . . . . . . . 10 (((AB) = r:(AB)–1-1-ontox) → (rA) Nc A)
30 ssun2 3427 . . . . . . . . . . . . . 14 B (AB)
31 f1ores 5300 . . . . . . . . . . . . . 14 ((r:(AB)–1-1x B (AB)) → (r B):B1-1-onto→(rB))
3215, 30, 31sylancl 643 . . . . . . . . . . . . 13 (r:(AB)–1-1-ontox → (r B):B1-1-onto→(rB))
33 f1ocnv 5299 . . . . . . . . . . . . 13 ((r B):B1-1-onto→(rB) → (r B):(rB)–1-1-ontoB)
34 ncdisjun.2 . . . . . . . . . . . . . . . 16 B V
3521, 34resex 5117 . . . . . . . . . . . . . . 15 (r B) V
3635cnvex 5102 . . . . . . . . . . . . . 14 (r B) V
3736f1oen 6033 . . . . . . . . . . . . 13 ((r B):(rB)–1-1-ontoB → (rB) ≈ B)
3832, 33, 373syl 18 . . . . . . . . . . . 12 (r:(AB)–1-1-ontox → (rB) ≈ B)
3938adantl 452 . . . . . . . . . . 11 (((AB) = r:(AB)–1-1-ontox) → (rB) ≈ B)
40 elnc 6125 . . . . . . . . . . 11 ((rB) Nc B ↔ (rB) ≈ B)
4139, 40sylibr 203 . . . . . . . . . 10 (((AB) = r:(AB)–1-1-ontox) → (rB) Nc B)
42 df-f1 4792 . . . . . . . . . . . . . 14 (r:(AB)–1-1x ↔ (r:(AB)–→x Fun r))
4342simprbi 450 . . . . . . . . . . . . 13 (r:(AB)–1-1x → Fun r)
44 imain 5172 . . . . . . . . . . . . 13 (Fun r → (r “ (AB)) = ((rA) ∩ (rB)))
4515, 43, 443syl 18 . . . . . . . . . . . 12 (r:(AB)–1-1-ontox → (r “ (AB)) = ((rA) ∩ (rB)))
4645adantl 452 . . . . . . . . . . 11 (((AB) = r:(AB)–1-1-ontox) → (r “ (AB)) = ((rA) ∩ (rB)))
47 imaeq2 4938 . . . . . . . . . . . . 13 ((AB) = → (r “ (AB)) = (r))
48 ima0 5013 . . . . . . . . . . . . 13 (r) =
4947, 48syl6eq 2401 . . . . . . . . . . . 12 ((AB) = → (r “ (AB)) = )
5049adantr 451 . . . . . . . . . . 11 (((AB) = r:(AB)–1-1-ontox) → (r “ (AB)) = )
5146, 50eqtr3d 2387 . . . . . . . . . 10 (((AB) = r:(AB)–1-1-ontox) → ((rA) ∩ (rB)) = )
52 eladdci 4399 . . . . . . . . . 10 (((rA) Nc A (rB) Nc B ((rA) ∩ (rB)) = ) → ((rA) ∪ (rB)) ( Nc A +c Nc B))
5329, 41, 51, 52syl3anc 1182 . . . . . . . . 9 (((AB) = r:(AB)–1-1-ontox) → ((rA) ∪ (rB)) ( Nc A +c Nc B))
5414, 53eqeltrrd 2428 . . . . . . . 8 (((AB) = r:(AB)–1-1-ontox) → x ( Nc A +c Nc B))
5554ex 423 . . . . . . 7 ((AB) = → (r:(AB)–1-1-ontoxx ( Nc A +c Nc B)))
563, 55syl5 28 . . . . . 6 ((AB) = → (r:x1-1-onto→(AB) → x ( Nc A +c Nc B)))
5756exlimdv 1636 . . . . 5 ((AB) = → (r r:x1-1-onto→(AB) → x ( Nc A +c Nc B)))
582, 57syl5bi 208 . . . 4 ((AB) = → (x ≈ (AB) → x ( Nc A +c Nc B)))
59 eladdc 4398 . . . . 5 (x ( Nc A +c Nc B) ↔ p Nc Aq Nc B((pq) = x = (pq)))
60 simplrl 736 . . . . . . . . . 10 ((((AB) = (p Nc A q Nc B)) (pq) = ) → p Nc A)
61 elnc 6125 . . . . . . . . . 10 (p Nc ApA)
6260, 61sylib 188 . . . . . . . . 9 ((((AB) = (p Nc A q Nc B)) (pq) = ) → pA)
63 simplrr 737 . . . . . . . . . 10 ((((AB) = (p Nc A q Nc B)) (pq) = ) → q Nc B)
64 elnc 6125 . . . . . . . . . 10 (q Nc BqB)
6563, 64sylib 188 . . . . . . . . 9 ((((AB) = (p Nc A q Nc B)) (pq) = ) → qB)
66 simpr 447 . . . . . . . . 9 ((((AB) = (p Nc A q Nc B)) (pq) = ) → (pq) = )
67 simpll 730 . . . . . . . . 9 ((((AB) = (p Nc A q Nc B)) (pq) = ) → (AB) = )
68 unen 6048 . . . . . . . . 9 (((pA qB) ((pq) = (AB) = )) → (pq) ≈ (AB))
6962, 65, 66, 67, 68syl22anc 1183 . . . . . . . 8 ((((AB) = (p Nc A q Nc B)) (pq) = ) → (pq) ≈ (AB))
70 breq1 4642 . . . . . . . 8 (x = (pq) → (x ≈ (AB) ↔ (pq) ≈ (AB)))
7169, 70syl5ibrcom 213 . . . . . . 7 ((((AB) = (p Nc A q Nc B)) (pq) = ) → (x = (pq) → x ≈ (AB)))
7271expimpd 586 . . . . . 6 (((AB) = (p Nc A q Nc B)) → (((pq) = x = (pq)) → x ≈ (AB)))
7372rexlimdvva 2745 . . . . 5 ((AB) = → (p Nc Aq Nc B((pq) = x = (pq)) → x ≈ (AB)))
7459, 73syl5bi 208 . . . 4 ((AB) = → (x ( Nc A +c Nc B) → x ≈ (AB)))
7558, 74impbid 183 . . 3 ((AB) = → (x ≈ (AB) ↔ x ( Nc A +c Nc B)))
761, 75syl5bb 248 . 2 ((AB) = → (x Nc (AB) ↔ x ( Nc A +c Nc B)))
7776eqrdv 2351 1 ((AB) = Nc (AB) = ( Nc A +c Nc B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550   +c cplc 4375   class class class wbr 4639   “ cima 4722  ◡ccnv 4771  dom cdm 4772  ran crn 4773   ↾ cres 4774  Fun wfun 4775  –→wf 4777  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780   ≈ cen 6028   Nc cnc 6091 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-en 6029  df-nc 6101 This theorem is referenced by:  ncaddccl  6144  1p1e2c  6155  2p1e3c  6156  dflec2  6210  addcdi  6250  nchoicelem7  6295  nchoicelem14  6302
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