Theorem fin23lem24 9144

Description: Lemma for fin23 9211. In a class of ordinals, each element is fully identified by those of its predecessors which also belong to the class. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Assertion

Ref	Expression
fin23lem24	|- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> ( ( C i^i B ) = ( D i^i B ) <-> C = D ) )

Proof of Theorem fin23lem24

Step	Hyp	Ref	Expression
1		simpll 790	. . . . . 6 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> Ord A )
2		simplr 792	. . . . . . 7 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> B C_ A )
3		simprl 794	. . . . . . 7 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> C e. B )
4	2, 3	sseldd 3604	. . . . . 6 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> C e. A )
5		ordelord 5745	. . . . . 6 |- ( ( Ord A /\ C e. A ) -> Ord C )
6	1, 4, 5	syl2anc 693	. . . . 5 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> Ord C )
7		simprr 796	. . . . . . 7 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> D e. B )
8	2, 7	sseldd 3604	. . . . . 6 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> D e. A )
9		ordelord 5745	. . . . . 6 |- ( ( Ord A /\ D e. A ) -> Ord D )
10	1, 8, 9	syl2anc 693	. . . . 5 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> Ord D )
11		ordtri3 5759	. . . . . 6 |- ( ( Ord C /\ Ord D ) -> ( C = D <-> -. ( C e. D \/ D e. C ) ) )
12	11	necon2abid 2836	. . . . 5 |- ( ( Ord C /\ Ord D ) -> ( ( C e. D \/ D e. C ) <-> C =/= D ) )
13	6, 10, 12	syl2anc 693	. . . 4 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> ( ( C e. D \/ D e. C ) <-> C =/= D ) )
14		simpr 477	. . . . . . . . 9 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ C e. D ) -> C e. D )
15		simplrl 800	. . . . . . . . 9 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ C e. D ) -> C e. B )
16	14, 15	elind 3798	. . . . . . . 8 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ C e. D ) -> C e. ( D i^i B ) )
17	6	adantr 481	. . . . . . . . . 10 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ C e. D ) -> Ord C )
18		ordirr 5741	. . . . . . . . . 10 |- ( Ord C -> -. C e. C )
19	17, 18	syl 17	. . . . . . . . 9 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ C e. D ) -> -. C e. C )
20		inss1 3833	. . . . . . . . . 10 |- ( C i^i B ) C_ C
21	20	sseli 3599	. . . . . . . . 9 |- ( C e. ( C i^i B ) -> C e. C )
22	19, 21	nsyl 135	. . . . . . . 8 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ C e. D ) -> -. C e. ( C i^i B ) )
23		nelne1 2890	. . . . . . . 8 |- ( ( C e. ( D i^i B ) /\ -. C e. ( C i^i B ) ) -> ( D i^i B ) =/= ( C i^i B ) )
24	16, 22, 23	syl2anc 693	. . . . . . 7 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ C e. D ) -> ( D i^i B ) =/= ( C i^i B ) )
25	24	necomd 2849	. . . . . 6 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ C e. D ) -> ( C i^i B ) =/= ( D i^i B ) )
26		simpr 477	. . . . . . . 8 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ D e. C ) -> D e. C )
27		simplrr 801	. . . . . . . 8 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ D e. C ) -> D e. B )
28	26, 27	elind 3798	. . . . . . 7 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ D e. C ) -> D e. ( C i^i B ) )
29	10	adantr 481	. . . . . . . . 9 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ D e. C ) -> Ord D )
30		ordirr 5741	. . . . . . . . 9 |- ( Ord D -> -. D e. D )
31	29, 30	syl 17	. . . . . . . 8 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ D e. C ) -> -. D e. D )
32		inss1 3833	. . . . . . . . 9 |- ( D i^i B ) C_ D
33	32	sseli 3599	. . . . . . . 8 |- ( D e. ( D i^i B ) -> D e. D )
34	31, 33	nsyl 135	. . . . . . 7 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ D e. C ) -> -. D e. ( D i^i B ) )
35		nelne1 2890	. . . . . . 7 |- ( ( D e. ( C i^i B ) /\ -. D e. ( D i^i B ) ) -> ( C i^i B ) =/= ( D i^i B ) )
36	28, 34, 35	syl2anc 693	. . . . . 6 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ D e. C ) -> ( C i^i B ) =/= ( D i^i B ) )
37	25, 36	jaodan 826	. . . . 5 |- ( ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) /\ ( C e. D \/ D e. C ) ) -> ( C i^i B ) =/= ( D i^i B ) )
38	37	ex 450	. . . 4 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> ( ( C e. D \/ D e. C ) -> ( C i^i B ) =/= ( D i^i B ) ) )
39	13, 38	sylbird 250	. . 3 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> ( C =/= D -> ( C i^i B ) =/= ( D i^i B ) ) )
40	39	necon4d 2818	. 2 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> ( ( C i^i B ) = ( D i^i B ) -> C = D ) )
41		ineq1 3807	. 2 |- ( C = D -> ( C i^i B ) = ( D i^i B ) )
42	40, 41	impbid1 215	1 |- ( ( ( Ord A /\ B C_ A ) /\ ( C e. B /\ D e. B ) ) -> ( ( C i^i B ) = ( D i^i B ) <-> C = D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   Ord word 5722

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by:  fin23lem23  9148