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Theorem bnj1098 28815
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 945 . . . . . . 7  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  <->  ( n  e.  D  /\  i  e.  n  /\  i  =/=  (/) ) )
2 df-3an 936 . . . . . . 7  |-  ( ( n  e.  D  /\  i  e.  n  /\  i  =/=  (/) )  <->  ( (
n  e.  D  /\  i  e.  n )  /\  i  =/=  (/) ) )
31, 2bitri 240 . . . . . 6  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  <->  ( (
n  e.  D  /\  i  e.  n )  /\  i  =/=  (/) ) )
4 simpr 447 . . . . . . . 8  |-  ( ( n  e.  D  /\  i  e.  n )  ->  i  e.  n )
5 bnj1098.1 . . . . . . . . . 10  |-  D  =  ( om  \  { (/)
} )
65bnj923 28798 . . . . . . . . 9  |-  ( n  e.  D  ->  n  e.  om )
76adantr 451 . . . . . . . 8  |-  ( ( n  e.  D  /\  i  e.  n )  ->  n  e.  om )
8 elnn 4666 . . . . . . . 8  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
94, 7, 8syl2anc 642 . . . . . . 7  |-  ( ( n  e.  D  /\  i  e.  n )  ->  i  e.  om )
109anim1i 551 . . . . . 6  |-  ( ( ( n  e.  D  /\  i  e.  n
)  /\  i  =/=  (/) )  ->  ( i  e.  om  /\  i  =/=  (/) ) )
113, 10sylbi 187 . . . . 5  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  (
i  e.  om  /\  i  =/=  (/) ) )
12 nnsuc 4673 . . . . 5  |-  ( ( i  e.  om  /\  i  =/=  (/) )  ->  E. j  e.  om  i  =  suc  j )
1311, 12syl 15 . . . 4  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  E. j  e.  om  i  =  suc  j )
14 df-rex 2549 . . . . . 6  |-  ( E. j  e.  om  i  =  suc  j  <->  E. j
( j  e.  om  /\  i  =  suc  j
) )
1514imbi2i 303 . . . . 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 1840 . . . . 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 243 . . . 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 199 . . 3  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  om  /\  i  =  suc  j ) )
19 ancr 532 . . 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 28749 . 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 2791 . . . . . 6  |-  j  e. 
_V
2221bnj216 28760 . . . . 5  |-  ( i  =  suc  j  -> 
j  e.  i )
2322ad2antlr 707 . . . 4  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
j  e.  i )
24 simpr2 962 . . . 4  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
i  e.  n )
25 3simpc 954 . . . . . . 7  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  (
i  e.  n  /\  n  e.  D )
)
2625ancomd 438 . . . . . 6  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  (
n  e.  D  /\  i  e.  n )
)
2726adantl 452 . . . . 5  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
( n  e.  D  /\  i  e.  n
) )
28 nnord 4664 . . . . . 6  |-  ( n  e.  om  ->  Ord  n )
297, 28syl 15 . . . . 5  |-  ( ( n  e.  D  /\  i  e.  n )  ->  Ord  n )
30 ordtr1 4435 . . . . 5  |-  ( Ord  n  ->  ( (
j  e.  i  /\  i  e.  n )  ->  j  e.  n ) )
3127, 29, 303syl 18 . . . 4  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
( ( j  e.  i  /\  i  e.  n )  ->  j  e.  n ) )
3223, 24, 31mp2and 660 . . 3  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
j  e.  n )
33 simplr 731 . . 3  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
i  =  suc  j
)
3432, 33jca 518 . 2  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
( j  e.  n  /\  i  =  suc  j ) )
3520, 34bnj1023 28812 1  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149   (/)c0 3455   {csn 3640   Ord word 4391   suc csuc 4394   omcom 4656
This theorem is referenced by:  bnj1110  29012  bnj1128  29020  bnj1145  29023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657
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