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Theorem evennn 4506
 Description: An even finite cardinal is a finite cardinal. (Contributed by SF, 20-Jan-2015.)
Assertion
Ref Expression
evennn (A EvenfinA Nn )

Proof of Theorem evennn
Dummy variables n x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . . . . . 6 (x = A → (x = (n +c n) ↔ A = (n +c n)))
21rexbidv 2635 . . . . 5 (x = A → (n Nn x = (n +c n) ↔ n Nn A = (n +c n)))
3 neeq1 2524 . . . . 5 (x = A → (xA))
42, 3anbi12d 691 . . . 4 (x = A → ((n Nn x = (n +c n) x) ↔ (n Nn A = (n +c n) A)))
5 df-evenfin 4444 . . . 4 Evenfin = {x (n Nn x = (n +c n) x)}
64, 5elab2g 2987 . . 3 (A Evenfin → (A Evenfin ↔ (n Nn A = (n +c n) A)))
76ibi 232 . 2 (A Evenfin → (n Nn A = (n +c n) A))
8 nncaddccl 4419 . . . . . 6 ((n Nn n Nn ) → (n +c n) Nn )
98anidms 626 . . . . 5 (n Nn → (n +c n) Nn )
10 eleq1a 2422 . . . . 5 ((n +c n) Nn → (A = (n +c n) → A Nn ))
119, 10syl 15 . . . 4 (n Nn → (A = (n +c n) → A Nn ))
1211rexlimiv 2732 . . 3 (n Nn A = (n +c n) → A Nn )
1312adantr 451 . 2 ((n Nn A = (n +c n) A) → A Nn )
147, 13syl 15 1 (A EvenfinA Nn )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550   Nn cnnc 4373   +c cplc 4375   Evenfin cevenfin 4436 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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-0c 4377  df-addc 4378  df-nnc 4379  df-evenfin 4444 This theorem is referenced by:  evenoddnnnul  4514
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