Theorem evennn 4506

Description: An even finite cardinal is a finite cardinal. (Contributed by SF, 20-Jan-2015.)

Assertion

Ref	Expression
evennn	⊢ (A ∈ Evenfin → A ∈ Nn )

Proof of Theorem evennn

Dummy variables n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . . . . 6 ⊢ (x = A → (x = (n +c n) ↔ A = (n +c n)))
2	1	rexbidv 2635	. . . . 5 ⊢ (x = A → (∃n ∈ Nn x = (n +c n) ↔ ∃n ∈ Nn A = (n +c n)))
3		neeq1 2524	. . . . 5 ⊢ (x = A → (x ≠ ∅ ↔ A ≠ ∅))
4	2, 3	anbi12d 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 ≠ ∅)}
6	4, 5	elab2g 2987	. . 3 ⊢ (A ∈ Evenfin → (A ∈ Evenfin ↔ (∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅)))
7	6	ibi 232	. 2 ⊢ (A ∈ Evenfin → (∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅))
8		nncaddccl 4419	. . . . . 6 ⊢ ((n ∈ Nn ∧ n ∈ Nn ) → (n +c n) ∈ Nn )
9	8	anidms 626	. . . . 5 ⊢ (n ∈ Nn → (n +c n) ∈ Nn )
10		eleq1a 2422	. . . . 5 ⊢ ((n +c n) ∈ Nn → (A = (n +c n) → A ∈ Nn ))
11	9, 10	syl 15	. . . 4 ⊢ (n ∈ Nn → (A = (n +c n) → A ∈ Nn ))
12	11	rexlimiv 2732	. . 3 ⊢ (∃n ∈ Nn A = (n +c n) → A ∈ Nn )
13	12	adantr 451	. 2 ⊢ ((∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅) → A ∈ Nn )
14	7, 13	syl 15	1 ⊢ (A ∈ Evenfin → A ∈ Nn )

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550   Nn cnnc 4373   +_c cplc 4375   Even_fin 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-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