Theorem oneo 7661

Description: If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)

Assertion

Ref	Expression
oneo	|- ( ( A e. On /\ B e. On /\ C = ( 2o .o A ) ) -> -. suc C = ( 2o .o B ) )

Proof of Theorem oneo

Step	Hyp	Ref	Expression
1		onnbtwn 5818	. . 3 |- ( A e. On -> -. ( A e. B /\ B e. suc A ) )
2	1	3ad2ant1 1082	. 2 |- ( ( A e. On /\ B e. On /\ C = ( 2o .o A ) ) -> -. ( A e. B /\ B e. suc A ) )
3		suceq 5790	. . . . 5 |- ( C = ( 2o .o A ) -> suc C = suc ( 2o .o A ) )
4	3	eqeq1d 2624	. . . 4 |- ( C = ( 2o .o A ) -> ( suc C = ( 2o .o B ) <-> suc ( 2o .o A ) = ( 2o .o B ) ) )
5	4	3ad2ant3 1084	. . 3 |- ( ( A e. On /\ B e. On /\ C = ( 2o .o A ) ) -> ( suc C = ( 2o .o B ) <-> suc ( 2o .o A ) = ( 2o .o B ) ) )
6		ovex 6678	. . . . . . . 8 |- ( 2o .o A ) e. _V
7	6	sucid 5804	. . . . . . 7 |- ( 2o .o A ) e. suc ( 2o .o A )
8		eleq2 2690	. . . . . . 7 |- ( suc ( 2o .o A ) = ( 2o .o B ) -> ( ( 2o .o A ) e. suc ( 2o .o A ) <-> ( 2o .o A ) e. ( 2o .o B ) ) )
9	7, 8	mpbii 223	. . . . . 6 |- ( suc ( 2o .o A ) = ( 2o .o B ) -> ( 2o .o A ) e. ( 2o .o B ) )
10		2on 7568	. . . . . . . 8 |- 2o e. On
11		omord 7648	. . . . . . . 8 |- ( ( A e. On /\ B e. On /\ 2o e. On ) -> ( ( A e. B /\ (/) e. 2o ) <-> ( 2o .o A ) e. ( 2o .o B ) ) )
12	10, 11	mp3an3 1413	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( ( A e. B /\ (/) e. 2o ) <-> ( 2o .o A ) e. ( 2o .o B ) ) )
13		simpl 473	. . . . . . 7 |- ( ( A e. B /\ (/) e. 2o ) -> A e. B )
14	12, 13	syl6bir 244	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( ( 2o .o A ) e. ( 2o .o B ) -> A e. B ) )
15	9, 14	syl5 34	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( suc ( 2o .o A ) = ( 2o .o B ) -> A e. B ) )
16		simpr 477	. . . . . . . . 9 |- ( ( ( A e. On /\ B e. On ) /\ suc ( 2o .o A ) = ( 2o .o B ) ) -> suc ( 2o .o A ) = ( 2o .o B ) )
17		omcl 7616	. . . . . . . . . . . . 13 |- ( ( 2o e. On /\ A e. On ) -> ( 2o .o A ) e. On )
18	10, 17	mpan 706	. . . . . . . . . . . 12 |- ( A e. On -> ( 2o .o A ) e. On )
19		oa1suc 7611	. . . . . . . . . . . 12 |- ( ( 2o .o A ) e. On -> ( ( 2o .o A ) +o 1o ) = suc ( 2o .o A ) )
20	18, 19	syl 17	. . . . . . . . . . 11 |- ( A e. On -> ( ( 2o .o A ) +o 1o ) = suc ( 2o .o A ) )
21		1on 7567	. . . . . . . . . . . . . . . 16 |- 1o e. On
22	21	elexi 3213	. . . . . . . . . . . . . . 15 |- 1o e. _V
23	22	sucid 5804	. . . . . . . . . . . . . 14 |- 1o e. suc 1o
24		df-2o 7561	. . . . . . . . . . . . . 14 |- 2o = suc 1o
25	23, 24	eleqtrri 2700	. . . . . . . . . . . . 13 |- 1o e. 2o
26		oaord 7627	. . . . . . . . . . . . . . 15 |- ( ( 1o e. On /\ 2o e. On /\ ( 2o .o A ) e. On ) -> ( 1o e. 2o <-> ( ( 2o .o A ) +o 1o ) e. ( ( 2o .o A ) +o 2o ) ) )
27	21, 10, 26	mp3an12 1414	. . . . . . . . . . . . . 14 |- ( ( 2o .o A ) e. On -> ( 1o e. 2o <-> ( ( 2o .o A ) +o 1o ) e. ( ( 2o .o A ) +o 2o ) ) )
28	18, 27	syl 17	. . . . . . . . . . . . 13 |- ( A e. On -> ( 1o e. 2o <-> ( ( 2o .o A ) +o 1o ) e. ( ( 2o .o A ) +o 2o ) ) )
29	25, 28	mpbii 223	. . . . . . . . . . . 12 |- ( A e. On -> ( ( 2o .o A ) +o 1o ) e. ( ( 2o .o A ) +o 2o ) )
30		omsuc 7606	. . . . . . . . . . . . 13 |- ( ( 2o e. On /\ A e. On ) -> ( 2o .o suc A ) = ( ( 2o .o A ) +o 2o ) )
31	10, 30	mpan 706	. . . . . . . . . . . 12 |- ( A e. On -> ( 2o .o suc A ) = ( ( 2o .o A ) +o 2o ) )
32	29, 31	eleqtrrd 2704	. . . . . . . . . . 11 |- ( A e. On -> ( ( 2o .o A ) +o 1o ) e. ( 2o .o suc A ) )
33	20, 32	eqeltrrd 2702	. . . . . . . . . 10 |- ( A e. On -> suc ( 2o .o A ) e. ( 2o .o suc A ) )
34	33	ad2antrr 762	. . . . . . . . 9 |- ( ( ( A e. On /\ B e. On ) /\ suc ( 2o .o A ) = ( 2o .o B ) ) -> suc ( 2o .o A ) e. ( 2o .o suc A ) )
35	16, 34	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( A e. On /\ B e. On ) /\ suc ( 2o .o A ) = ( 2o .o B ) ) -> ( 2o .o B ) e. ( 2o .o suc A ) )
36		suceloni 7013	. . . . . . . . . . 11 |- ( A e. On -> suc A e. On )
37		omord 7648	. . . . . . . . . . . 12 |- ( ( B e. On /\ suc A e. On /\ 2o e. On ) -> ( ( B e. suc A /\ (/) e. 2o ) <-> ( 2o .o B ) e. ( 2o .o suc A ) ) )
38	10, 37	mp3an3 1413	. . . . . . . . . . 11 |- ( ( B e. On /\ suc A e. On ) -> ( ( B e. suc A /\ (/) e. 2o ) <-> ( 2o .o B ) e. ( 2o .o suc A ) ) )
39	36, 38	sylan2 491	. . . . . . . . . 10 |- ( ( B e. On /\ A e. On ) -> ( ( B e. suc A /\ (/) e. 2o ) <-> ( 2o .o B ) e. ( 2o .o suc A ) ) )
40	39	ancoms 469	. . . . . . . . 9 |- ( ( A e. On /\ B e. On ) -> ( ( B e. suc A /\ (/) e. 2o ) <-> ( 2o .o B ) e. ( 2o .o suc A ) ) )
41	40	adantr 481	. . . . . . . 8 |- ( ( ( A e. On /\ B e. On ) /\ suc ( 2o .o A ) = ( 2o .o B ) ) -> ( ( B e. suc A /\ (/) e. 2o ) <-> ( 2o .o B ) e. ( 2o .o suc A ) ) )
42	35, 41	mpbird 247	. . . . . . 7 |- ( ( ( A e. On /\ B e. On ) /\ suc ( 2o .o A ) = ( 2o .o B ) ) -> ( B e. suc A /\ (/) e. 2o ) )
43	42	simpld 475	. . . . . 6 |- ( ( ( A e. On /\ B e. On ) /\ suc ( 2o .o A ) = ( 2o .o B ) ) -> B e. suc A )
44	43	ex 450	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( suc ( 2o .o A ) = ( 2o .o B ) -> B e. suc A ) )
45	15, 44	jcad 555	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( suc ( 2o .o A ) = ( 2o .o B ) -> ( A e. B /\ B e. suc A ) ) )
46	45	3adant3 1081	. . 3 |- ( ( A e. On /\ B e. On /\ C = ( 2o .o A ) ) -> ( suc ( 2o .o A ) = ( 2o .o B ) -> ( A e. B /\ B e. suc A ) ) )
47	5, 46	sylbid 230	. 2 |- ( ( A e. On /\ B e. On /\ C = ( 2o .o A ) ) -> ( suc C = ( 2o .o B ) -> ( A e. B /\ B e. suc A ) ) )
48	2, 47	mtod 189	1 |- ( ( A e. On /\ B e. On /\ C = ( 2o .o A ) ) -> -. suc C = ( 2o .o B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   (/)c0 3915   Oncon0 5723   suc csuc 5725  (class class class)co 6650   1oc1o 7553   2oc2o 7554   +o coa 7557   .o comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565

This theorem is referenced by: (None)