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Theorem bnj1098 28872
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 947 . . . . . . 7  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  <->  ( n  e.  D  /\  i  e.  n  /\  i  =/=  (/) ) )
2 df-3an 938 . . . . . . 7  |-  ( ( n  e.  D  /\  i  e.  n  /\  i  =/=  (/) )  <->  ( (
n  e.  D  /\  i  e.  n )  /\  i  =/=  (/) ) )
31, 2bitri 241 . . . . . 6  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  <->  ( (
n  e.  D  /\  i  e.  n )  /\  i  =/=  (/) ) )
4 simpr 448 . . . . . . . 8  |-  ( ( n  e.  D  /\  i  e.  n )  ->  i  e.  n )
5 bnj1098.1 . . . . . . . . . 10  |-  D  =  ( om  \  { (/)
} )
65bnj923 28855 . . . . . . . . 9  |-  ( n  e.  D  ->  n  e.  om )
76adantr 452 . . . . . . . 8  |-  ( ( n  e.  D  /\  i  e.  n )  ->  n  e.  om )
8 elnn 4822 . . . . . . . 8  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
94, 7, 8syl2anc 643 . . . . . . 7  |-  ( ( n  e.  D  /\  i  e.  n )  ->  i  e.  om )
109anim1i 552 . . . . . 6  |-  ( ( ( n  e.  D  /\  i  e.  n
)  /\  i  =/=  (/) )  ->  ( i  e.  om  /\  i  =/=  (/) ) )
113, 10sylbi 188 . . . . 5  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  (
i  e.  om  /\  i  =/=  (/) ) )
12 nnsuc 4829 . . . . 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 2680 . . . . . 6  |-  ( E. j  e.  om  i  =  suc  j  <->  E. j
( j  e.  om  /\  i  =  suc  j
) )
1514imbi2i 304 . . . . 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 1918 . . . . 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 244 . . . 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 200 . . 3  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  om  /\  i  =  suc  j ) )
19 ancr 533 . . 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 28806 . 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 2927 . . . . . 6  |-  j  e. 
_V
2221bnj216 28817 . . . . 5  |-  ( i  =  suc  j  -> 
j  e.  i )
2322ad2antlr 708 . . . 4  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
j  e.  i )
24 simpr2 964 . . . 4  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
i  e.  n )
25 3simpc 956 . . . . . . 7  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  (
i  e.  n  /\  n  e.  D )
)
2625ancomd 439 . . . . . 6  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  (
n  e.  D  /\  i  e.  n )
)
2726adantl 453 . . . . 5  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
( n  e.  D  /\  i  e.  n
) )
28 nnord 4820 . . . . . 6  |-  ( n  e.  om  ->  Ord  n )
297, 28syl 16 . . . . 5  |-  ( ( n  e.  D  /\  i  e.  n )  ->  Ord  n )
30 ordtr1 4592 . . . . 5  |-  ( Ord  n  ->  ( (
j  e.  i  /\  i  e.  n )  ->  j  e.  n ) )
3127, 29, 303syl 19 . . . 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 661 . . 3  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
j  e.  n )
33 simplr 732 . . 3  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
i  =  suc  j
)
3432, 33jca 519 . 2  |-  ( ( ( j  e.  om  /\  i  =  suc  j
)  /\  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )  -> 
( j  e.  n  /\  i  =  suc  j ) )
3520, 34bnj1023 28869 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 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675    \ cdif 3285   (/)c0 3596   {csn 3782   Ord word 4548   suc csuc 4551   omcom 4812
This theorem is referenced by:  bnj1110  29069  bnj1128  29077  bnj1145  29080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813
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