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Theorem phplem3 7695
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 4944 . 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 7694 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
52enref 7545 . . . 4  |-  A  ~~  A
6 nnord 6686 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
7 orddif 4971 . . . . . 6  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
86, 7syl 16 . . . . 5  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
9 sneq 4037 . . . . . . 7  |-  ( A  =  B  ->  { A }  =  { B } )
109difeq2d 3622 . . . . . 6  |-  ( A  =  B  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
1110eqcoms 2479 . . . . 5  |-  ( B  =  A  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
128, 11sylan9eq 2528 . . . 4  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  =  ( suc 
A  \  { B } ) )
135, 12syl5breq 4482 . . 3  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
144, 13jaodan 783 . 2  |-  ( ( A  e.  om  /\  ( B  e.  A  \/  B  =  A
) )  ->  A  ~~  ( suc  A  \  { B } ) )
151, 14sylan2 474 1  |-  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473   {csn 4027   class class class wbr 4447   Ord word 4877   suc csuc 4880   omcom 6678    ~~ cen 7510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-om 6679  df-en 7514
This theorem is referenced by:  phplem4  7696  php  7698
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