MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phplem3 Unicode version

Theorem phplem3 7042
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.)
Hypotheses
Ref Expression
phplem2.1  |-  A  e. 
_V
phplem2.2  |-  B  e. 
_V
Assertion
Ref Expression
phplem3  |-  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )

Proof of Theorem phplem3
StepHypRef Expression
1 elsuci 4458 . 2  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
2 phplem2.1 . . . 4  |-  A  e. 
_V
3 phplem2.2 . . . 4  |-  B  e. 
_V
42, 3phplem2 7041 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
52enref 6894 . . . 4  |-  A  ~~  A
6 nnord 4664 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
7 orddif 4486 . . . . . 6  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
86, 7syl 15 . . . . 5  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
9 sneq 3651 . . . . . . 7  |-  ( A  =  B  ->  { A }  =  { B } )
109difeq2d 3294 . . . . . 6  |-  ( A  =  B  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
1110eqcoms 2286 . . . . 5  |-  ( B  =  A  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
128, 11sylan9eq 2335 . . . 4  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  =  ( suc 
A  \  { B } ) )
135, 12syl5breq 4058 . . 3  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
144, 13jaodan 760 . 2  |-  ( ( A  e.  om  /\  ( B  e.  A  \/  B  =  A
) )  ->  A  ~~  ( suc  A  \  { B } ) )
151, 14sylan2 460 1  |-  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   {csn 3640   class class class wbr 4023   Ord word 4391   suc csuc 4394   omcom 4656    ~~ cen 6860
This theorem is referenced by:  phplem4  7043  php  7045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-en 6864
  Copyright terms: Public domain W3C validator