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Theorem phplem3 7753
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 5489 . 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 7752 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
52enref 7602 . . . 4  |-  A  ~~  A
6 nnord 6700 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
7 orddif 5516 . . . . . 6  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
86, 7syl 17 . . . . 5  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
9 sneq 3978 . . . . . . 7  |-  ( A  =  B  ->  { A }  =  { B } )
109difeq2d 3551 . . . . . 6  |-  ( A  =  B  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
1110eqcoms 2459 . . . . 5  |-  ( B  =  A  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
128, 11sylan9eq 2505 . . . 4  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  =  ( suc 
A  \  { B } ) )
135, 12syl5breq 4438 . . 3  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
144, 13jaodan 794 . 2  |-  ( ( A  e.  om  /\  ( B  e.  A  \/  B  =  A
) )  ->  A  ~~  ( suc  A  \  { B } ) )
151, 14sylan2 477 1  |-  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    \ cdif 3401   {csn 3968   class class class wbr 4402   Ord word 5422   suc csuc 5425   omcom 6692    ~~ cen 7566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-om 6693  df-en 7570
This theorem is referenced by:  phplem4  7754  php  7756
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