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Theorem bnj158 31816
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 2507 . . 3  |-  ( m  e.  D  <->  m  e.  ( om  \  { (/) } ) )
3 eldifsn 4021 . . 3  |-  ( m  e.  ( om  \  { (/)
} )  <->  ( m  e.  om  /\  m  =/=  (/) ) )
42, 3bitri 249 . 2  |-  ( m  e.  D  <->  ( m  e.  om  /\  m  =/=  (/) ) )
5 nnsuc 6514 . 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 369    = wceq 1369    e. wcel 1756    =/= wne 2620   E.wrex 2737    \ cdif 3346   (/)c0 3658   {csn 3898   suc csuc 4742   omcom 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-tr 4407  df-eprel 4653  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-om 6498
This theorem is referenced by:  bnj168  31817  bnj600  32008  bnj986  32043
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