Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj158 Structured version   Unicode version

Theorem bnj158 28997
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 2499 . . 3  |-  ( m  e.  D  <->  m  e.  ( om  \  { (/) } ) )
3 eldifsn 3919 . . 3  |-  ( m  e.  ( om  \  { (/)
} )  <->  ( m  e.  om  /\  m  =/=  (/) ) )
42, 3bitri 241 . 2  |-  ( m  e.  D  <->  ( m  e.  om  /\  m  =/=  (/) ) )
5 nnsuc 4854 . 2  |-  ( ( m  e.  om  /\  m  =/=  (/) )  ->  E. p  e.  om  m  =  suc  p )
64, 5sylbi 188 1  |-  ( m  e.  D  ->  E. p  e.  om  m  =  suc  p )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309   (/)c0 3620   {csn 3806   suc csuc 4575   omcom 4837
This theorem is referenced by:  bnj168  28998  bnj600  29191  bnj986  29226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838
  Copyright terms: Public domain W3C validator