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Theorem bnj1098 28560
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 28543 . . . . . . . . 9  |-  ( n  e.  D  ->  n  e.  om )
76adantr 451 . . . . . . . 8  |-  ( ( n  e.  D  /\  i  e.  n )  ->  n  e.  om )
8 elnn 4745 . . . . . . . 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 4752 . . . . 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 2625 . . . . . 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 1904 . . . . 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 28494 . 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 2867 . . . . . 6  |-  j  e. 
_V
2221bnj216 28505 . . . . 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 4743 . . . . . 6  |-  ( n  e.  om  ->  Ord  n )
297, 28syl 15 . . . . 5  |-  ( ( n  e.  D  /\  i  e.  n )  ->  Ord  n )
30 ordtr1 4514 . . . . 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 28557 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 1541    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620    \ cdif 3225   (/)c0 3531   {csn 3716   Ord word 4470   suc csuc 4473   omcom 4735
This theorem is referenced by:  bnj1110  28757  bnj1128  28765  bnj1145  28768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-tr 4193  df-eprel 4384  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736
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