Theorem eqncg 6127

Description: Equality of cardinalities. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
eqncg	⊢ (A ∈ V → ( Nc A = Nc B ↔ A ≈ B))

Proof of Theorem eqncg

Step	Hyp	Ref	Expression
1		ncidg 6123	. . . . . . 7 ⊢ (A ∈ V → A ∈ Nc A)
2	1	adantr 451	. . . . . 6 ⊢ ((A ∈ V ∧ Nc A = Nc B) → A ∈ Nc A)
3		eleq2 2414	. . . . . . 7 ⊢ ( Nc A = Nc B → (A ∈ Nc A ↔ A ∈ Nc B))
4	3	adantl 452	. . . . . 6 ⊢ ((A ∈ V ∧ Nc A = Nc B) → (A ∈ Nc A ↔ A ∈ Nc B))
5	2, 4	mpbid 201	. . . . 5 ⊢ ((A ∈ V ∧ Nc A = Nc B) → A ∈ Nc B)
6		df-nc 6102	. . . . 5 ⊢ Nc B = [B] ≈
7	5, 6	syl6eleq 2443	. . . 4 ⊢ ((A ∈ V ∧ Nc A = Nc B) → A ∈ [B] ≈ )
8		ecexr 5951	. . . 4 ⊢ (A ∈ [B] ≈ → B ∈ V)
9	7, 8	syl 15	. . 3 ⊢ ((A ∈ V ∧ Nc A = Nc B) → B ∈ V)
10	9	ex 423	. 2 ⊢ (A ∈ V → ( Nc A = Nc B → B ∈ V))
11		brex 4690	. . . 4 ⊢ (A ≈ B → (A ∈ V ∧ B ∈ V))
12	11	simprd 449	. . 3 ⊢ (A ≈ B → B ∈ V)
13	12	a1i 10	. 2 ⊢ (A ∈ V → (A ≈ B → B ∈ V))
14		ener 6040	. . . . . 6 ⊢ ≈ Er V
15	14	a1i 10	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ≈ Er V)
16		dmen 6042	. . . . . 6 ⊢ dom ≈ = V
17	16	a1i 10	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → dom ≈ = V)
18		elex 2868	. . . . . 6 ⊢ (A ∈ V → A ∈ V)
19	18	adantr 451	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → A ∈ V)
20		simpr 447	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → B ∈ V)
21	15, 17, 19, 20	erth 5969	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → (A ≈ B ↔ [A] ≈ = [B] ≈ ))
22		df-nc 6102	. . . . 5 ⊢ Nc A = [A] ≈
23	22, 6	eqeq12i 2366	. . . 4 ⊢ ( Nc A = Nc B ↔ [A] ≈ = [B] ≈ )
24	21, 23	syl6rbbr 255	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → ( Nc A = Nc B ↔ A ≈ B))
25	24	ex 423	. 2 ⊢ (A ∈ V → (B ∈ V → ( Nc A = Nc B ↔ A ≈ B)))
26	10, 13, 25	pm5.21ndd 343	1 ⊢ (A ∈ V → ( Nc A = Nc B ↔ A ≈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   class class class wbr 4640  dom cdm 4773   Er cer 5899  [cec 5946   ≈ cen 6029   Nc cnc 6092

This theorem was proved from 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 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 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 referenced by:  eqnc  6128  ncpw1pwneg  6202  canncb  6333