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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
StepHypRef 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  =/=  (/) ) )
31, 2bitri 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  \  { (/)
} )
65bnj923 32058 . . . . . . . . 9  |-  ( n  e.  D  ->  n  e.  om )
76adantr 465 . . . . . . . 8  |-  ( ( n  e.  D  /\  i  e.  n )  ->  n  e.  om )
8 elnn 6583 . . . . . . . 8  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
94, 7, 8syl2anc 661 . . . . . . 7  |-  ( ( n  e.  D  /\  i  e.  n )  ->  i  e.  om )
109anim1i 568 . . . . . 6  |-  ( ( ( n  e.  D  /\  i  e.  n
)  /\  i  =/=  (/) )  ->  ( i  e.  om  /\  i  =/=  (/) ) )
113, 10sylbi 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 )
1311, 12syl 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
) )
1514imbi2i 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
) ) )
1715, 16bitr4i 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 ) ) )
1813, 17mpbi 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 ) ) ) )
2018, 19bnj101 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
2221bnj216 32020 . . . . 5  |-  ( i  =  suc  j  -> 
j  e.  i )
2322ad2antlr 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 )
)
2625ancomd 451 . . . . . 6  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  (
n  e.  D  /\  i  e.  n )
)
2726adantl 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 ) )
3027, 7, 28, 294syl 21 . . . 4  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
( ( j  e.  i  /\  i  e.  n )  ->  j  e.  n ) )
3123, 24, 30mp2and 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
)
3331, 32jca 532 . 2  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
( j  e.  n  /\  i  =  suc  j ) )
3420, 33bnj1023 32071 1  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
Colors of variables: wff setvar class
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
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