Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj986 Structured version   Unicode version

Theorem bnj986 33745
Description: Technical lemma for bnj69 33799. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj986.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj986.10  |-  D  =  ( om  \  { (/)
} )
bnj986.15  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
Assertion
Ref Expression
bnj986  |-  ( ch 
->  E. m E. p ta )
Distinct variable group:    m, n, p
Allowed substitution hints:    ph( f, m, n, p)    ps( f, m, n, p)    ch( f, m, n, p)    ta( f, m, n, p)    D( f, m, n, p)

Proof of Theorem bnj986
StepHypRef Expression
1 bnj986.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj986.10 . . . . . . 7  |-  D  =  ( om  \  { (/)
} )
32bnj158 33517 . . . . . 6  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
41, 3bnj769 33553 . . . . 5  |-  ( ch 
->  E. m  e.  om  n  =  suc  m )
54bnj1196 33586 . . . 4  |-  ( ch 
->  E. m ( m  e.  om  /\  n  =  suc  m ) )
6 vex 3098 . . . . . 6  |-  n  e. 
_V
76sucex 6631 . . . . 5  |-  suc  n  e.  _V
87isseti 3101 . . . 4  |-  E. p  p  =  suc  n
95, 8jctir 538 . . 3  |-  ( ch 
->  ( E. m ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
10 exdistr 1762 . . . 4  |-  ( E. m E. p ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n )  <->  E. m
( ( m  e. 
om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
11 19.41v 1757 . . . 4  |-  ( E. m ( ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n )  <->  ( E. m ( m  e. 
om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
1210, 11bitr2i 250 . . 3  |-  ( ( E. m ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n )  <->  E. m E. p ( ( m  e.  om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
139, 12sylib 196 . 2  |-  ( ch 
->  E. m E. p
( ( m  e. 
om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
14 bnj986.15 . . . 4  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
15 df-3an 976 . . . 4  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  <-> 
( ( m  e. 
om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
1614, 15bitri 249 . . 3  |-  ( ta  <->  ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
17162exbii 1655 . 2  |-  ( E. m E. p ta  <->  E. m E. p ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
1813, 17sylibr 212 1  |-  ( ch 
->  E. m E. p ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383   E.wex 1599    e. wcel 1804   E.wrex 2794    \ cdif 3458   (/)c0 3770   {csn 4014   suc csuc 4870    Fn wfn 5573   omcom 6685    /\ w-bnj17 33471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-om 6686  df-bnj17 33472
This theorem is referenced by:  bnj996  33746
  Copyright terms: Public domain W3C validator