Theorem bnj1098 33327

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 984	. . . . . . 7 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) <-> ( n e. D /\ i e. n /\ i =/= (/) ) )
2		df-3an 975	. . . . . . 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 33311	. . . . . . . . 9 |- ( n e. D -> n e. om )
7	6	adantr 465	. . . . . . . 8 |- ( ( n e. D /\ i e. n ) -> n e. om )
8		elnn 6705	. . . . . . . 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 6712	. . . . 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 2823	. . . . . 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 1943	. . . . 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 33262	. 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 3121	. . . . . 6 |- j e. _V
22	21	bnj216 33273	. . . . 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 1003	. . . 4 |- ( ( ( j e. om /\ i = suc j ) /\ ( i =/= (/) /\ i e. n /\ n e. D ) ) -> i e. n )
25		3simpc 995	. . . . . . 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 6703	. . . . 5 |- ( n e. om -> Ord n )
29		ordtr1 4927	. . . . 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 33324	1 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   E.wrex 2818   \ cdif 3478   (/)c0 3790   {csn 4033   Ord word 4883   suc csuc 4886   omcom 6695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-om 6696

This theorem is referenced by:  bnj1110  33523  bnj1128  33531  bnj1145  33534