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Theorem phplem3 7280
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 4639 . 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 7279 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
52enref 7132 . . . 4  |-  A  ~~  A
6 nnord 4845 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
7 orddif 4667 . . . . . 6  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
86, 7syl 16 . . . . 5  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
9 sneq 3817 . . . . . . 7  |-  ( A  =  B  ->  { A }  =  { B } )
109difeq2d 3457 . . . . . 6  |-  ( A  =  B  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
1110eqcoms 2438 . . . . 5  |-  ( B  =  A  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
128, 11sylan9eq 2487 . . . 4  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  =  ( suc 
A  \  { B } ) )
135, 12syl5breq 4239 . . 3  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
144, 13jaodan 761 . 2  |-  ( ( A  e.  om  /\  ( B  e.  A  \/  B  =  A
) )  ->  A  ~~  ( suc  A  \  { B } ) )
151, 14sylan2 461 1  |-  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309   {csn 3806   class class class wbr 4204   Ord word 4572   suc csuc 4575   omcom 4837    ~~ cen 7098
This theorem is referenced by:  phplem4  7281  php  7283
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-pow 4369  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-id 4490  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  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-en 7102
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