Theorem cantnflem2 8587

Description: Lemma for cantnf 8590. (Contributed by Mario Carneiro, 28-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
oemapval.t	|- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }
cantnf.c	|- ( ph -> C e. ( A ^o B ) )
cantnf.s	|- ( ph -> C C_ ran ( A CNF B ) )
cantnf.e	|- ( ph -> (/) e. C )

Assertion

Ref	Expression
cantnflem2	|- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) )

Distinct variable groups:   x, w, y, z, B   w, C, x, y, z   w, A, x, y, z   x, S, y, z   ph, x, y, z

Allowed substitution hints:   ph( w)   S( w)   T( x, y, z, w)

Proof of Theorem cantnflem2

Step	Hyp	Ref	Expression
1		cantnfs.a	. . 3 |- ( ph -> A e. On )
2		cantnfs.b	. . . . . . . . . 10 |- ( ph -> B e. On )
3		oecl 7617	. . . . . . . . . 10 |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On )
4	1, 2, 3	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( A ^o B ) e. On )
5		cantnf.c	. . . . . . . . 9 |- ( ph -> C e. ( A ^o B ) )
6		onelon 5748	. . . . . . . . 9 |- ( ( ( A ^o B ) e. On /\ C e. ( A ^o B ) ) -> C e. On )
7	4, 5, 6	syl2anc 693	. . . . . . . 8 |- ( ph -> C e. On )
8		cantnf.e	. . . . . . . 8 |- ( ph -> (/) e. C )
9		ondif1 7581	. . . . . . . 8 |- ( C e. ( On \ 1o ) <-> ( C e. On /\ (/) e. C ) )
10	7, 8, 9	sylanbrc 698	. . . . . . 7 |- ( ph -> C e. ( On \ 1o ) )
11	10	eldifbd 3587	. . . . . 6 |- ( ph -> -. C e. 1o )
12		ssel 3597	. . . . . . 7 |- ( ( A ^o B ) C_ 1o -> ( C e. ( A ^o B ) -> C e. 1o ) )
13	5, 12	syl5com 31	. . . . . 6 |- ( ph -> ( ( A ^o B ) C_ 1o -> C e. 1o ) )
14	11, 13	mtod 189	. . . . 5 |- ( ph -> -. ( A ^o B ) C_ 1o )
15		oe0m 7598	. . . . . . . . 9 |- ( B e. On -> ( (/) ^o B ) = ( 1o \ B ) )
16	2, 15	syl 17	. . . . . . . 8 |- ( ph -> ( (/) ^o B ) = ( 1o \ B ) )
17		difss 3737	. . . . . . . 8 |- ( 1o \ B ) C_ 1o
18	16, 17	syl6eqss 3655	. . . . . . 7 |- ( ph -> ( (/) ^o B ) C_ 1o )
19		oveq1 6657	. . . . . . . 8 |- ( A = (/) -> ( A ^o B ) = ( (/) ^o B ) )
20	19	sseq1d 3632	. . . . . . 7 |- ( A = (/) -> ( ( A ^o B ) C_ 1o <-> ( (/) ^o B ) C_ 1o ) )
21	18, 20	syl5ibrcom 237	. . . . . 6 |- ( ph -> ( A = (/) -> ( A ^o B ) C_ 1o ) )
22		oe1m 7625	. . . . . . . 8 |- ( B e. On -> ( 1o ^o B ) = 1o )
23		eqimss 3657	. . . . . . . 8 |- ( ( 1o ^o B ) = 1o -> ( 1o ^o B ) C_ 1o )
24	2, 22, 23	3syl 18	. . . . . . 7 |- ( ph -> ( 1o ^o B ) C_ 1o )
25		oveq1 6657	. . . . . . . 8 |- ( A = 1o -> ( A ^o B ) = ( 1o ^o B ) )
26	25	sseq1d 3632	. . . . . . 7 |- ( A = 1o -> ( ( A ^o B ) C_ 1o <-> ( 1o ^o B ) C_ 1o ) )
27	24, 26	syl5ibrcom 237	. . . . . 6 |- ( ph -> ( A = 1o -> ( A ^o B ) C_ 1o ) )
28	21, 27	jaod 395	. . . . 5 |- ( ph -> ( ( A = (/) \/ A = 1o ) -> ( A ^o B ) C_ 1o ) )
29	14, 28	mtod 189	. . . 4 |- ( ph -> -. ( A = (/) \/ A = 1o ) )
30		elpri 4197	. . . . 5 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
31		df2o3 7573	. . . . 5 |- 2o = { (/) , 1o }
32	30, 31	eleq2s 2719	. . . 4 |- ( A e. 2o -> ( A = (/) \/ A = 1o ) )
33	29, 32	nsyl 135	. . 3 |- ( ph -> -. A e. 2o )
34	1, 33	eldifd 3585	. 2 |- ( ph -> A e. ( On \ 2o ) )
35	34, 10	jca 554	1 |- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   (/)c0 3915   {cpr 4179   {copab 4712   dom cdm 5114   ran crn 5115   Oncon0 5723   ` cfv 5888  (class class class)co 6650   1oc1o 7553   2oc2o 7554   ^o coe 7559   CNF ccnf 8558

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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  cantnflem3  8588  cantnflem4  8589