Theorem sucoddeven 4511

Description: The successor of an odd natural is even. (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
sucoddeven	|- ((A e. Oddfin /\ (A +o 1c) =/= (/)) -> (A +o 1c) e. Evenfin )

Proof of Theorem sucoddeven

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

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . . . . . . 8 |- (x = A -> (x = ((n +o n) +o 1c) <-> A = ((n +o n) +o 1c)))
2	1	rexbidv 2635	. . . . . . 7 |- (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	. . . . . . 7 |- (x = A -> (x =/= (/) <-> A =/= (/)))
4	2, 3	anbi12d 691	. . . . . 6 |- (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	. . . . . 6 |- Oddfin = {x | (E.n e. Nn x = ((n +o n) +o 1c) /\ x =/= (/))}
6	4, 5	elab2g 2987	. . . . 5 |- (A e. Oddfin -> (A e. Oddfin <-> (E.n e. Nn A = ((n +o n) +o 1c) /\ A =/= (/))))
7	6	ibi 232	. . . 4 |- (A e. Oddfin -> (E.n e. Nn A = ((n +o n) +o 1c) /\ A =/= (/)))
8		peano2 4403	. . . . . . . 8 |- (n e. Nn -> (n +o 1c) e. Nn )
9		addc32 4416	. . . . . . . . . . 11 |- ((n +o n) +o 1c) = ((n +o 1c) +o n)
10	9	addceq1i 4386	. . . . . . . . . 10 |- (((n +o n) +o 1c) +o 1c) = (((n +o 1c) +o n) +o 1c)
11		addcass 4415	. . . . . . . . . 10 |- (((n +o 1c) +o n) +o 1c) = ((n +o 1c) +o (n +o 1c))
12	10, 11	eqtri 2373	. . . . . . . . 9 |- (((n +o n) +o 1c) +o 1c) = ((n +o 1c) +o (n +o 1c))
13		addceq12 4385	. . . . . . . . . . . 12 |- ((m = (n +o 1c) /\ m = (n +o 1c)) -> (m +o m) = ((n +o 1c) +o (n +o 1c)))
14	13	anidms 626	. . . . . . . . . . 11 |- (m = (n +o 1c) -> (m +o m) = ((n +o 1c) +o (n +o 1c)))
15	14	eqeq2d 2364	. . . . . . . . . 10 |- (m = (n +o 1c) -> ((((n +o n) +o 1c) +o 1c) = (m +o m) <-> (((n +o n) +o 1c) +o 1c) = ((n +o 1c) +o (n +o 1c))))
16	15	rspcev 2955	. . . . . . . . 9 |- (((n +o 1c) e. Nn /\ (((n +o n) +o 1c) +o 1c) = ((n +o 1c) +o (n +o 1c))) -> E.m e. Nn (((n +o n) +o 1c) +o 1c) = (m +o m))
17	12, 16	mpan2 652	. . . . . . . 8 |- ((n +o 1c) e. Nn -> E.m e. Nn (((n +o n) +o 1c) +o 1c) = (m +o m))
18	8, 17	syl 15	. . . . . . 7 |- (n e. Nn -> E.m e. Nn (((n +o n) +o 1c) +o 1c) = (m +o m))
19		addceq1 4383	. . . . . . . . . . 11 |- (A = ((n +o n) +o 1c) -> (A +o 1c) = (((n +o n) +o 1c) +o 1c))
20	19	eqeq1d 2361	. . . . . . . . . 10 |- (A = ((n +o n) +o 1c) -> ((A +o 1c) = (m +o m) <-> (((n +o n) +o 1c) +o 1c) = (m +o m)))
21	20	rexbidv 2635	. . . . . . . . 9 |- (A = ((n +o n) +o 1c) -> (E.m e. Nn (A +o 1c) = (m +o m) <-> E.m e. Nn (((n +o n) +o 1c) +o 1c) = (m +o m)))
22	21	biimprd 214	. . . . . . . 8 |- (A = ((n +o n) +o 1c) -> (E.m e. Nn (((n +o n) +o 1c) +o 1c) = (m +o m) -> E.m e. Nn (A +o 1c) = (m +o m)))
23	22	com12 27	. . . . . . 7 |- (E.m e. Nn (((n +o n) +o 1c) +o 1c) = (m +o m) -> (A = ((n +o n) +o 1c) -> E.m e. Nn (A +o 1c) = (m +o m)))
24	18, 23	syl 15	. . . . . 6 |- (n e. Nn -> (A = ((n +o n) +o 1c) -> E.m e. Nn (A +o 1c) = (m +o m)))
25	24	rexlimiv 2732	. . . . 5 |- (E.n e. Nn A = ((n +o n) +o 1c) -> E.m e. Nn (A +o 1c) = (m +o m))
26	25	adantr 451	. . . 4 |- ((E.n e. Nn A = ((n +o n) +o 1c) /\ A =/= (/)) -> E.m e. Nn (A +o 1c) = (m +o m))
27	7, 26	syl 15	. . 3 |- (A e. Oddfin -> E.m e. Nn (A +o 1c) = (m +o m))
28	27	anim1i 551	. 2 |- ((A e. Oddfin /\ (A +o 1c) =/= (/)) -> (E.m e. Nn (A +o 1c) = (m +o m) /\ (A +o 1c) =/= (/)))
29		1cex 4142	. . . . 5 |- 1c e. _V
30		addcexg 4393	. . . . 5 |- ((A e. Oddfin /\ 1c e. _V) -> (A +o 1c) e. _V)
31	29, 30	mpan2 652	. . . 4 |- (A e. Oddfin -> (A +o 1c) e. _V)
32		eqeq1 2359	. . . . . . 7 |- (x = (A +o 1c) -> (x = (m +o m) <-> (A +o 1c) = (m +o m)))
33	32	rexbidv 2635	. . . . . 6 |- (x = (A +o 1c) -> (E.m e. Nn x = (m +o m) <-> E.m e. Nn (A +o 1c) = (m +o m)))
34		neeq1 2524	. . . . . 6 |- (x = (A +o 1c) -> (x =/= (/) <-> (A +o 1c) =/= (/)))
35	33, 34	anbi12d 691	. . . . 5 |- (x = (A +o 1c) -> ((E.m e. Nn x = (m +o m) /\ x =/= (/)) <-> (E.m e. Nn (A +o 1c) = (m +o m) /\ (A +o 1c) =/= (/))))
36		df-evenfin 4444	. . . . 5 |- Evenfin = {x | (E.m e. Nn x = (m +o m) /\ x =/= (/))}
37	35, 36	elab2g 2987	. . . 4 |- ((A +o 1c) e. _V -> ((A +o 1c) e. Evenfin <-> (E.m e. Nn (A +o 1c) = (m +o m) /\ (A +o 1c) =/= (/))))
38	31, 37	syl 15	. . 3 |- (A e. Oddfin -> ((A +o 1c) e. Evenfin <-> (E.m e. Nn (A +o 1c) = (m +o m) /\ (A +o 1c) =/= (/))))
39	38	adantr 451	. 2 |- ((A e. Oddfin /\ (A +o 1c) =/= (/)) -> ((A +o 1c) e. Evenfin <-> (E.m e. Nn (A +o 1c) = (m +o m) /\ (A +o 1c) =/= (/))))
40	28, 39	mpbird 223	1 |- ((A e. Oddfin /\ (A +o 1c) =/= (/)) -> (A +o 1c) e. Evenfin )

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2516  E.wrex 2615  _Vcvv 2859  (/)c0 3550  1_cc1c 4134   Nn cnnc 4373   +o cplc 4375   Even_fin cevenfin 4436   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-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-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193  df-addc 4378  df-nnc 4379  df-evenfin 4444  df-oddfin 4445

This theorem is referenced by:  evenoddnnnul  4514  vinf  4555