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Theorem bnj158 34185
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj158.1  |-  D  =  ( om  \  { (/)
} )
Assertion
Ref Expression
bnj158  |-  ( m  e.  D  ->  E. p  e.  om  m  =  suc  p )
Distinct variable group:    m, p
Allowed substitution hints:    D( m, p)

Proof of Theorem bnj158
StepHypRef Expression
1 bnj158.1 . . . 4  |-  D  =  ( om  \  { (/)
} )
21eleq2i 2532 . . 3  |-  ( m  e.  D  <->  m  e.  ( om  \  { (/) } ) )
3 eldifsn 4141 . . 3  |-  ( m  e.  ( om  \  { (/)
} )  <->  ( m  e.  om  /\  m  =/=  (/) ) )
42, 3bitri 249 . 2  |-  ( m  e.  D  <->  ( m  e.  om  /\  m  =/=  (/) ) )
5 nnsuc 6690 . 2  |-  ( ( m  e.  om  /\  m  =/=  (/) )  ->  E. p  e.  om  m  =  suc  p )
64, 5sylbi 195 1  |-  ( m  e.  D  ->  E. p  e.  om  m  =  suc  p )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    \ cdif 3458   (/)c0 3783   {csn 4016   suc csuc 4869   omcom 6673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-om 6674
This theorem is referenced by:  bnj168  34186  bnj600  34378  bnj986  34413
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