New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  eqncg GIF version

Theorem eqncg 6126
 Description: Equality of cardinalities. (Contributed by SF, 24-Feb-2015.)
Assertion
Ref Expression
eqncg (A V → ( Nc A = Nc BAB))

Proof of Theorem eqncg
StepHypRef Expression
1 ncidg 6122 . . . . . . 7 (A VA Nc A)
21adantr 451 . . . . . 6 ((A V Nc A = Nc B) → A Nc A)
3 eleq2 2414 . . . . . . 7 ( Nc A = Nc B → (A Nc AA Nc B))
43adantl 452 . . . . . 6 ((A V Nc A = Nc B) → (A Nc AA Nc B))
52, 4mpbid 201 . . . . 5 ((A V Nc A = Nc B) → A Nc B)
6 df-nc 6101 . . . . 5 Nc B = [B] ≈
75, 6syl6eleq 2443 . . . 4 ((A V Nc A = Nc B) → A [B] ≈ )
8 ecexr 5950 . . . 4 (A [B] ≈ → B V)
97, 8syl 15 . . 3 ((A V Nc A = Nc B) → B V)
109ex 423 . 2 (A V → ( Nc A = Nc BB V))
11 brex 4689 . . . 4 (AB → (A V B V))
1211simprd 449 . . 3 (ABB V)
1312a1i 10 . 2 (A V → (ABB V))
14 ener 6039 . . . . . 6 Er V
1514a1i 10 . . . . 5 ((A V B V) → ≈ Er V)
16 dmen 6041 . . . . . 6 dom ≈ = V
1716a1i 10 . . . . 5 ((A V B V) → dom ≈ = V)
18 elex 2867 . . . . . 6 (A VA V)
1918adantr 451 . . . . 5 ((A V B V) → A V)
20 simpr 447 . . . . 5 ((A V B V) → B V)
2115, 17, 19, 20erth 5968 . . . 4 ((A V B V) → (AB ↔ [A] ≈ = [B] ≈ ))
22 df-nc 6101 . . . . 5 Nc A = [A] ≈
2322, 6eqeq12i 2366 . . . 4 ( Nc A = Nc B ↔ [A] ≈ = [B] ≈ )
2421, 23syl6rbbr 255 . . 3 ((A V B V) → ( Nc A = Nc BAB))
2524ex 423 . 2 (A V → (B V → ( Nc A = Nc BAB)))
2610, 13, 25pm5.21ndd 343 1 (A V → ( Nc A = Nc BAB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   class class class wbr 4639  dom cdm 4772   Er cer 5898  [cec 5945   ≈ 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:  eqnc  6127  ncpw1pwneg  6201  canncb  6332
 Copyright terms: Public domain W3C validator