Theorem fin23lem23 9148

Description: Lemma for fin23lem22 9149. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Assertion

Ref	Expression
fin23lem23	|- ( ( ( S C_ om /\ -. S e. Fin ) /\ i e. om ) -> E! j e. S ( j i^i S ) ~~ i )

Distinct variable group:   i, j, S

Proof of Theorem fin23lem23

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem26 9147	. 2 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ i e. om ) -> E. j e. S ( j i^i S ) ~~ i )
2		ensym 8005	. . . . . 6 |- ( ( a i^i S ) ~~ i -> i ~~ ( a i^i S ) )
3		entr 8008	. . . . . 6 |- ( ( ( j i^i S ) ~~ i /\ i ~~ ( a i^i S ) ) -> ( j i^i S ) ~~ ( a i^i S ) )
4	2, 3	sylan2 491	. . . . 5 |- ( ( ( j i^i S ) ~~ i /\ ( a i^i S ) ~~ i ) -> ( j i^i S ) ~~ ( a i^i S ) )
5		simpl 473	. . . . . . . . 9 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> S C_ om )
6		simprl 794	. . . . . . . . 9 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> j e. S )
7	5, 6	sseldd 3604	. . . . . . . 8 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> j e. om )
8		nnfi 8153	. . . . . . . . 9 |- ( j e. om -> j e. Fin )
9		inss1 3833	. . . . . . . . 9 |- ( j i^i S ) C_ j
10		ssfi 8180	. . . . . . . . 9 |- ( ( j e. Fin /\ ( j i^i S ) C_ j ) -> ( j i^i S ) e. Fin )
11	8, 9, 10	sylancl 694	. . . . . . . 8 |- ( j e. om -> ( j i^i S ) e. Fin )
12	7, 11	syl 17	. . . . . . 7 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> ( j i^i S ) e. Fin )
13		simprr 796	. . . . . . . . 9 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> a e. S )
14	5, 13	sseldd 3604	. . . . . . . 8 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> a e. om )
15		nnfi 8153	. . . . . . . . 9 |- ( a e. om -> a e. Fin )
16		inss1 3833	. . . . . . . . 9 |- ( a i^i S ) C_ a
17		ssfi 8180	. . . . . . . . 9 |- ( ( a e. Fin /\ ( a i^i S ) C_ a ) -> ( a i^i S ) e. Fin )
18	15, 16, 17	sylancl 694	. . . . . . . 8 |- ( a e. om -> ( a i^i S ) e. Fin )
19	14, 18	syl 17	. . . . . . 7 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> ( a i^i S ) e. Fin )
20		nnord 7073	. . . . . . . . . 10 |- ( j e. om -> Ord j )
21		nnord 7073	. . . . . . . . . 10 |- ( a e. om -> Ord a )
22		ordtri2or2 5823	. . . . . . . . . 10 |- ( ( Ord j /\ Ord a ) -> ( j C_ a \/ a C_ j ) )
23	20, 21, 22	syl2an 494	. . . . . . . . 9 |- ( ( j e. om /\ a e. om ) -> ( j C_ a \/ a C_ j ) )
24	7, 14, 23	syl2anc 693	. . . . . . . 8 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> ( j C_ a \/ a C_ j ) )
25		ssrin 3838	. . . . . . . . 9 |- ( j C_ a -> ( j i^i S ) C_ ( a i^i S ) )
26		ssrin 3838	. . . . . . . . 9 |- ( a C_ j -> ( a i^i S ) C_ ( j i^i S ) )
27	25, 26	orim12i 538	. . . . . . . 8 |- ( ( j C_ a \/ a C_ j ) -> ( ( j i^i S ) C_ ( a i^i S ) \/ ( a i^i S ) C_ ( j i^i S ) ) )
28	24, 27	syl 17	. . . . . . 7 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> ( ( j i^i S ) C_ ( a i^i S ) \/ ( a i^i S ) C_ ( j i^i S ) ) )
29		fin23lem25 9146	. . . . . . 7 |- ( ( ( j i^i S ) e. Fin /\ ( a i^i S ) e. Fin /\ ( ( j i^i S ) C_ ( a i^i S ) \/ ( a i^i S ) C_ ( j i^i S ) ) ) -> ( ( j i^i S ) ~~ ( a i^i S ) <-> ( j i^i S ) = ( a i^i S ) ) )
30	12, 19, 28, 29	syl3anc 1326	. . . . . 6 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> ( ( j i^i S ) ~~ ( a i^i S ) <-> ( j i^i S ) = ( a i^i S ) ) )
31		ordom 7074	. . . . . . 7 |- Ord om
32		fin23lem24 9144	. . . . . . 7 |- ( ( ( Ord om /\ S C_ om ) /\ ( j e. S /\ a e. S ) ) -> ( ( j i^i S ) = ( a i^i S ) <-> j = a ) )
33	31, 32	mpanl1 716	. . . . . 6 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> ( ( j i^i S ) = ( a i^i S ) <-> j = a ) )
34	30, 33	bitrd 268	. . . . 5 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> ( ( j i^i S ) ~~ ( a i^i S ) <-> j = a ) )
35	4, 34	syl5ib 234	. . . 4 |- ( ( S C_ om /\ ( j e. S /\ a e. S ) ) -> ( ( ( j i^i S ) ~~ i /\ ( a i^i S ) ~~ i ) -> j = a ) )
36	35	ralrimivva 2971	. . 3 |- ( S C_ om -> A. j e. S A. a e. S ( ( ( j i^i S ) ~~ i /\ ( a i^i S ) ~~ i ) -> j = a ) )
37	36	ad2antrr 762	. 2 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ i e. om ) -> A. j e. S A. a e. S ( ( ( j i^i S ) ~~ i /\ ( a i^i S ) ~~ i ) -> j = a ) )
38		ineq1 3807	. . . 4 |- ( j = a -> ( j i^i S ) = ( a i^i S ) )
39	38	breq1d 4663	. . 3 |- ( j = a -> ( ( j i^i S ) ~~ i <-> ( a i^i S ) ~~ i ) )
40	39	reu4 3400	. 2 |- ( E! j e. S ( j i^i S ) ~~ i <-> ( E. j e. S ( j i^i S ) ~~ i /\ A. j e. S A. a e. S ( ( ( j i^i S ) ~~ i /\ ( a i^i S ) ~~ i ) -> j = a ) ) )
41	1, 37, 40	sylanbrc 698	1 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ i e. om ) -> E! j e. S ( j i^i S ) ~~ i )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   i^i cin 3573   C_ wss 3574   class class class wbr 4653   Ord word 5722   omcom 7065   ~~ cen 7952   Fincfn 7955

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-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-rmo 2920  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-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  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-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-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  fin23lem22  9149  fin23lem27  9150