Theorem oddnn 4507

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

Assertion

Ref	Expression
oddnn	|- (A e. Oddfin -> A e. Nn )

Proof of Theorem oddnn

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 +o n) +o 1c) <-> A = ((n +o n) +o 1c)))
2	1	rexbidv 2635	. . . . 5 |- (x = A -> (E.n e. Nn x = ((n +o n) +o 1c) <-> E.n e. Nn A = ((n +o n) +o 1c)))
3		neeq1 2524	. . . . 5 |- (x = A -> (x =/= (/) <-> A =/= (/)))
4	2, 3	anbi12d 691	. . . 4 |- (x = A -> ((E.n e. Nn x = ((n +o n) +o 1c) /\ x =/= (/)) <-> (E.n e. Nn A = ((n +o n) +o 1c) /\ A =/= (/))))
5		df-oddfin 4445	. . . 4 |- Oddfin = {x | (E.n e. Nn x = ((n +o n) +o 1c) /\ x =/= (/))}
6	4, 5	elab2g 2987	. . 3 |- (A e. Oddfin -> (A e. Oddfin <-> (E.n e. Nn A = ((n +o n) +o 1c) /\ A =/= (/))))
7	6	ibi 232	. 2 |- (A e. Oddfin -> (E.n e. Nn A = ((n +o n) +o 1c) /\ A =/= (/)))
8		nncaddccl 4419	. . . . . 6 |- ((n e. Nn /\ n e. Nn ) -> (n +o n) e. Nn )
9	8	anidms 626	. . . . 5 |- (n e. Nn -> (n +o n) e. Nn )
10		peano2 4403	. . . . 5 |- ((n +o n) e. Nn -> ((n +o n) +o 1c) e. Nn )
11		eleq1a 2422	. . . . 5 |- (((n +o n) +o 1c) e. Nn -> (A = ((n +o n) +o 1c) -> A e. Nn ))
12	9, 10, 11	3syl 18	. . . 4 |- (n e. Nn -> (A = ((n +o n) +o 1c) -> A e. Nn ))
13	12	rexlimiv 2732	. . 3 |- (E.n e. Nn A = ((n +o n) +o 1c) -> A e. Nn )
14	13	adantr 451	. 2 |- ((E.n e. Nn A = ((n +o n) +o 1c) /\ A =/= (/)) -> A e. Nn )
15	7, 14	syl 15	1 |- (A e. Oddfin -> A e. Nn )

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2516  E.wrex 2615  (/)c0 3550  1_cc1c 4134   Nn cnnc 4373   +o cplc 4375   Odd_fin coddfin 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 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-oddfin 4445

This theorem is referenced by:  evenoddnnnul  4514