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Theorem phplem3 6853
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6855. (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 4403 . 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 6852 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
52enref 6764 . . . 4  |-  A  ~~  A
6 nnord 4611 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
7 orddif 4546 . . . . . 6  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
86, 7syl 14 . . . . 5  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
9 sneq 3603 . . . . . . 7  |-  ( A  =  B  ->  { A }  =  { B } )
109difeq2d 3253 . . . . . 6  |-  ( A  =  B  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
1110eqcoms 2180 . . . . 5  |-  ( B  =  A  ->  ( suc  A  \  { A } )  =  ( suc  A  \  { B } ) )
128, 11sylan9eq 2230 . . . 4  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  =  ( suc 
A  \  { B } ) )
135, 12breqtrid 4040 . . 3  |-  ( ( A  e.  om  /\  B  =  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
144, 13jaodan 797 . 2  |-  ( ( A  e.  om  /\  ( B  e.  A  \/  B  =  A
) )  ->  A  ~~  ( suc  A  \  { B } ) )
151, 14sylan2 286 1  |-  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708    = wceq 1353    e. wcel 2148   _Vcvv 2737    \ cdif 3126   {csn 3592   class class class wbr 4003   Ord word 4362   suc csuc 4365   omcom 4589    ~~ cen 6737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-en 6740
This theorem is referenced by:  phplem4  6854  phplem3g  6855
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