Theorem bnj1098 30854

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1098.1	|- D = ( om \ { (/) } )

Assertion

Ref	Expression
bnj1098	|- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )

Distinct variable groups:   D, j   i, j   j, n

Allowed substitution hints:   D( i, n)

Proof of Theorem bnj1098

Step	Hyp	Ref	Expression
1		3anrev 1049	. . . . . . 7 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) <-> ( n e. D /\ i e. n /\ i =/= (/) ) )
2		df-3an 1039	. . . . . . 7 |- ( ( n e. D /\ i e. n /\ i =/= (/) ) <-> ( ( n e. D /\ i e. n ) /\ i =/= (/) ) )
3	1, 2	bitri 264	. . . . . 6 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) <-> ( ( n e. D /\ i e. n ) /\ i =/= (/) ) )
4		simpr 477	. . . . . . . 8 |- ( ( n e. D /\ i e. n ) -> i e. n )
5		bnj1098.1	. . . . . . . . . 10 |- D = ( om \ { (/) } )
6	5	bnj923 30838	. . . . . . . . 9 |- ( n e. D -> n e. om )
7	6	adantr 481	. . . . . . . 8 |- ( ( n e. D /\ i e. n ) -> n e. om )
8		elnn 7075	. . . . . . . 8 |- ( ( i e. n /\ n e. om ) -> i e. om )
9	4, 7, 8	syl2anc 693	. . . . . . 7 |- ( ( n e. D /\ i e. n ) -> i e. om )
10	9	anim1i 592	. . . . . 6 |- ( ( ( n e. D /\ i e. n ) /\ i =/= (/) ) -> ( i e. om /\ i =/= (/) ) )
11	3, 10	sylbi 207	. . . . 5 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( i e. om /\ i =/= (/) ) )
12		nnsuc 7082	. . . . 5 |- ( ( i e. om /\ i =/= (/) ) -> E. j e. om i = suc j )
13	11, 12	syl 17	. . . 4 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) -> E. j e. om i = suc j )
14		df-rex 2918	. . . . . 6 |- ( E. j e. om i = suc j <-> E. j ( j e. om /\ i = suc j ) )
15	14	imbi2i 326	. . . . 5 |- ( ( ( i =/= (/) /\ i e. n /\ n e. D ) -> E. j e. om i = suc j ) <-> ( ( i =/= (/) /\ i e. n /\ n e. D ) -> E. j ( j e. om /\ i = suc j ) ) )
16		19.37v 1910	. . . . 5 |- ( E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. om /\ i = suc j ) ) <-> ( ( i =/= (/) /\ i e. n /\ n e. D ) -> E. j ( j e. om /\ i = suc j ) ) )
17	15, 16	bitr4i 267	. . . 4 |- ( ( ( i =/= (/) /\ i e. n /\ n e. D ) -> E. j e. om i = suc j ) <-> E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. om /\ i = suc j ) ) )
18	13, 17	mpbi 220	. . 3 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. om /\ i = suc j ) )
19		ancr 572	. . 3 |- ( ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. om /\ i = suc j ) ) -> ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) ) )
20	18, 19	bnj101 30789	. 2 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) )
21		vex 3203	. . . . . 6 |- j e. _V
22	21	bnj216 30800	. . . . 5 |- ( i = suc j -> j e. i )
23	22	ad2antlr 763	. . . 4 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> j e. i )
24		simpr2 1068	. . . 4 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> i e. n )
25		3simpc 1060	. . . . . . 7 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( i e. n /\ n e. D ) )
26	25	ancomd 467	. . . . . 6 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( n e. D /\ i e. n ) )
27	26	adantl 482	. . . . 5 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> ( n e. D /\ i e. n ) )
28		nnord 7073	. . . . 5 |- ( n e. om -> Ord n )
29		ordtr1 5767	. . . . 5 |- ( Ord n -> ( ( j e. i /\ i e. n ) -> j e. n ) )
30	27, 7, 28, 29	4syl 19	. . . 4 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> ( ( j e. i /\ i e. n ) -> j e. n ) )
31	23, 24, 30	mp2and 715	. . 3 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> j e. n )
32		simplr 792	. . 3 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> i = suc j )
33	31, 32	jca 554	. 2 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> ( j e. n /\ i = suc j ) )
34	20, 33	bnj1023 30851	1 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   (/)c0 3915   {csn 4177   Ord word 5722   suc csuc 5725   omcom 7065

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-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-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-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  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by:  bnj1110  31050  bnj1128  31058  bnj1145  31061