New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  0ceven GIF version

Theorem 0ceven 4505
 Description: Cardinal zero is even. (Contributed by SF, 20-Jan-2015.)
Assertion
Ref Expression
0ceven 0c Evenfin

Proof of Theorem 0ceven
Dummy variables x n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano1 4402 . . 3 0c Nn
2 addcid2 4407 . . . 4 (0c +c 0c) = 0c
32eqcomi 2357 . . 3 0c = (0c +c 0c)
4 addceq12 4385 . . . . . 6 ((n = 0c n = 0c) → (n +c n) = (0c +c 0c))
54anidms 626 . . . . 5 (n = 0c → (n +c n) = (0c +c 0c))
65eqeq2d 2364 . . . 4 (n = 0c → (0c = (n +c n) ↔ 0c = (0c +c 0c)))
76rspcev 2955 . . 3 ((0c Nn 0c = (0c +c 0c)) → n Nn 0c = (n +c n))
81, 3, 7mp2an 653 . 2 n Nn 0c = (n +c n)
9 0ex 4110 . . . . 5 V
109snid 3760 . . . 4 {}
11 df-0c 4377 . . . 4 0c = {}
1210, 11eleqtrri 2426 . . 3 0c
13 ne0i 3556 . . 3 ( 0c → 0c)
1412, 13ax-mp 5 . 2 0c
15 0cex 4392 . . 3 0c V
16 eqeq1 2359 . . . . 5 (x = 0c → (x = (n +c n) ↔ 0c = (n +c n)))
1716rexbidv 2635 . . . 4 (x = 0c → (n Nn x = (n +c n) ↔ n Nn 0c = (n +c n)))
18 neeq1 2524 . . . 4 (x = 0c → (x ↔ 0c))
1917, 18anbi12d 691 . . 3 (x = 0c → ((n Nn x = (n +c n) x) ↔ (n Nn 0c = (n +c n) 0c)))
20 df-evenfin 4444 . . 3 Evenfin = {x (n Nn x = (n +c n) x)}
2115, 19, 20elab2 2988 . 2 (0c Evenfin ↔ (n Nn 0c = (n +c n) 0c))
228, 14, 21mpbir2an 886 1 0c Evenfin
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  {csn 3737   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   Evenfin cevenfin 4436 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  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-pw 3724  df-sn 3741  df-pr 3742  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193  df-0c 4377  df-addc 4378  df-nnc 4379  df-evenfin 4444 This theorem is referenced by:  evenoddnnnul  4514
 Copyright terms: Public domain W3C validator