Theorem bnj1098 32074

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 976	. . . . . . 7 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) <-> ( n e. D /\ i e. n /\ i =/= (/) ) )
2		df-3an 967	. . . . . . 7 |- ( ( n e. D /\ i e. n /\ i =/= (/) ) <-> ( ( n e. D /\ i e. n ) /\ i =/= (/) ) )
3	1, 2	bitri 249	. . . . . 6 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) <-> ( ( n e. D /\ i e. n ) /\ i =/= (/) ) )
4		simpr 461	. . . . . . . 8 |- ( ( n e. D /\ i e. n ) -> i e. n )
5		bnj1098.1	. . . . . . . . . 10 |- D = ( om \ { (/) } )
6	5	bnj923 32058	. . . . . . . . 9 |- ( n e. D -> n e. om )
7	6	adantr 465	. . . . . . . 8 |- ( ( n e. D /\ i e. n ) -> n e. om )
8		elnn 6583	. . . . . . . 8 |- ( ( i e. n /\ n e. om ) -> i e. om )
9	4, 7, 8	syl2anc 661	. . . . . . 7 |- ( ( n e. D /\ i e. n ) -> i e. om )
10	9	anim1i 568	. . . . . 6 |- ( ( ( n e. D /\ i e. n ) /\ i =/= (/) ) -> ( i e. om /\ i =/= (/) ) )
11	3, 10	sylbi 195	. . . . 5 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( i e. om /\ i =/= (/) ) )
12		nnsuc 6590	. . . . 5 |- ( ( i e. om /\ i =/= (/) ) -> E. j e. om i = suc j )
13	11, 12	syl 16	. . . 4 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) -> E. j e. om i = suc j )
14		df-rex 2799	. . . . . 6 |- ( E. j e. om i = suc j <-> E. j ( j e. om /\ i = suc j ) )
15	14	imbi2i 312	. . . . 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 1927	. . . . 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 252	. . . 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 208	. . 3 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. om /\ i = suc j ) )
19		ancr 549	. . 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 32009	. 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 3068	. . . . . 6 |- j e. _V
22	21	bnj216 32020	. . . . 5 |- ( i = suc j -> j e. i )
23	22	ad2antlr 726	. . . 4 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> j e. i )
24		simpr2 995	. . . 4 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> i e. n )
25		3simpc 987	. . . . . . 7 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( i e. n /\ n e. D ) )
26	25	ancomd 451	. . . . . 6 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( n e. D /\ i e. n ) )
27	26	adantl 466	. . . . 5 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> ( n e. D /\ i e. n ) )
28		nnord 6581	. . . . 5 |- ( n e. om -> Ord n )
29		ordtr1 4857	. . . . 5 |- ( Ord n -> ( ( j e. i /\ i e. n ) -> j e. n ) )
30	27, 7, 28, 29	4syl 21	. . . 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 679	. . 3 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> j e. n )
32		simplr 754	. . 3 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> i = suc j )
33	31, 32	jca 532	. 2 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> ( j e. n /\ i = suc j ) )
34	20, 33	bnj1023 32071	1 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2642   E.wrex 2794   \ cdif 3420   (/)c0 3732   {csn 3972   Ord word 4813   suc csuc 4816   omcom 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-tr 4481  df-eprel 4727  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-om 6574

This theorem is referenced by:  bnj1110  32270  bnj1128  32278  bnj1145  32281