Theorem 0ceven 4506

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.

Step	Hyp	Ref	Expression
1		peano1 4403	. . 3 ⊢ 0c ∈ Nn
2		addcid2 4408	. . . 4 ⊢ (0c +c 0c) = 0c
3	2	eqcomi 2357	. . 3 ⊢ 0c = (0c +c 0c)
4		addceq12 4386	. . . . . 6 ⊢ ((n = 0c ∧ n = 0c) → (n +c n) = (0c +c 0c))
5	4	anidms 626	. . . . 5 ⊢ (n = 0c → (n +c n) = (0c +c 0c))
6	5	eqeq2d 2364	. . . 4 ⊢ (n = 0c → (0c = (n +c n) ↔ 0c = (0c +c 0c)))
7	6	rspcev 2956	. . 3 ⊢ ((0c ∈ Nn ∧ 0c = (0c +c 0c)) → ∃n ∈ Nn 0c = (n +c n))
8	1, 3, 7	mp2an 653	. 2 ⊢ ∃n ∈ Nn 0c = (n +c n)
9		0ex 4111	. . . . 5 ⊢ ∅ ∈ V
10	9	snid 3761	. . . 4 ⊢ ∅ ∈ {∅}
11		df-0c 4378	. . . 4 ⊢ 0c = {∅}
12	10, 11	eleqtrri 2426	. . 3 ⊢ ∅ ∈ 0c
13		ne0i 3557	. . 3 ⊢ (∅ ∈ 0c → 0c ≠ ∅)
14	12, 13	ax-mp 5	. 2 ⊢ 0c ≠ ∅
15		0cex 4393	. . 3 ⊢ 0c ∈ V
16		eqeq1 2359	. . . . 5 ⊢ (x = 0c → (x = (n +c n) ↔ 0c = (n +c n)))
17	16	rexbidv 2636	. . . 4 ⊢ (x = 0c → (∃n ∈ Nn x = (n +c n) ↔ ∃n ∈ Nn 0c = (n +c n)))
18		neeq1 2525	. . . 4 ⊢ (x = 0c → (x ≠ ∅ ↔ 0c ≠ ∅))
19	17, 18	anbi12d 691	. . 3 ⊢ (x = 0c → ((∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅) ↔ (∃n ∈ Nn 0c = (n +c n) ∧ 0c ≠ ∅)))
20		df-evenfin 4445	. . 3 ⊢ Evenfin = {x ∣ (∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅)}
21	15, 19, 20	elab2 2989	. 2 ⊢ (0c ∈ Evenfin ↔ (∃n ∈ Nn 0c = (n +c n) ∧ 0c ≠ ∅))
22	8, 14, 21	mpbir2an 886	1 ⊢ 0c ∈ Evenfin

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  ∅c0 3551  {csn 3738   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   Even_fin cevenfin 4437

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194  df-0c 4378  df-addc 4379  df-nnc 4380  df-evenfin 4445

This theorem is referenced by:  evenoddnnnul  4515