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Theorem phplem2 7246
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
Hypotheses
Ref Expression
phplem2.1  |-  A  e. 
_V
phplem2.2  |-  B  e. 
_V
Assertion
Ref Expression
phplem2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )

Proof of Theorem phplem2
StepHypRef Expression
1 snex 4365 . . . . . 6  |-  { <. B ,  A >. }  e.  _V
2 phplem2.2 . . . . . . 7  |-  B  e. 
_V
3 phplem2.1 . . . . . . 7  |-  A  e. 
_V
42, 3f1osn 5674 . . . . . 6  |-  { <. B ,  A >. } : { B } -1-1-onto-> { A }
5 f1oen3g 7082 . . . . . 6  |-  ( ( { <. B ,  A >. }  e.  _V  /\  {
<. B ,  A >. } : { B } -1-1-onto-> { A } )  ->  { B }  ~~  { A }
)
61, 4, 5mp2an 654 . . . . 5  |-  { B }  ~~  { A }
7 difss 3434 . . . . . . 7  |-  ( A 
\  { B }
)  C_  A
83, 7ssexi 4308 . . . . . 6  |-  ( A 
\  { B }
)  e.  _V
98enref 7099 . . . . 5  |-  ( A 
\  { B }
)  ~~  ( A  \  { B } )
106, 9pm3.2i 442 . . . 4  |-  ( { B }  ~~  { A }  /\  ( A  \  { B }
)  ~~  ( A  \  { B } ) )
11 incom 3493 . . . . . 6  |-  ( { A }  i^i  ( A  \  { B }
) )  =  ( ( A  \  { B } )  i^i  { A } )
12 ssrin 3526 . . . . . . . . 9  |-  ( ( A  \  { B } )  C_  A  ->  ( ( A  \  { B } )  i^i 
{ A } ) 
C_  ( A  i^i  { A } ) )
137, 12ax-mp 8 . . . . . . . 8  |-  ( ( A  \  { B } )  i^i  { A } )  C_  ( A  i^i  { A }
)
14 nnord 4812 . . . . . . . . 9  |-  ( A  e.  om  ->  Ord  A )
15 orddisj 4579 . . . . . . . . 9  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
1614, 15syl 16 . . . . . . . 8  |-  ( A  e.  om  ->  ( A  i^i  { A }
)  =  (/) )
1713, 16syl5sseq 3356 . . . . . . 7  |-  ( A  e.  om  ->  (
( A  \  { B } )  i^i  { A } )  C_  (/) )
18 ss0 3618 . . . . . . 7  |-  ( ( ( A  \  { B } )  i^i  { A } )  C_  (/)  ->  (
( A  \  { B } )  i^i  { A } )  =  (/) )
1917, 18syl 16 . . . . . 6  |-  ( A  e.  om  ->  (
( A  \  { B } )  i^i  { A } )  =  (/) )
2011, 19syl5eq 2448 . . . . 5  |-  ( A  e.  om  ->  ( { A }  i^i  ( A  \  { B }
) )  =  (/) )
21 disjdif 3660 . . . . 5  |-  ( { B }  i^i  ( A  \  { B }
) )  =  (/)
2220, 21jctil 524 . . . 4  |-  ( A  e.  om  ->  (
( { B }  i^i  ( A  \  { B } ) )  =  (/)  /\  ( { A }  i^i  ( A  \  { B } ) )  =  (/) ) )
23 unen 7148 . . . 4  |-  ( ( ( { B }  ~~  { A }  /\  ( A  \  { B } )  ~~  ( A  \  { B }
) )  /\  (
( { B }  i^i  ( A  \  { B } ) )  =  (/)  /\  ( { A }  i^i  ( A  \  { B } ) )  =  (/) ) )  -> 
( { B }  u.  ( A  \  { B } ) )  ~~  ( { A }  u.  ( A  \  { B } ) ) )
2410, 22, 23sylancr 645 . . 3  |-  ( A  e.  om  ->  ( { B }  u.  ( A  \  { B }
) )  ~~  ( { A }  u.  ( A  \  { B }
) ) )
2524adantr 452 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { B }  u.  ( A  \  { B } ) )  ~~  ( { A }  u.  ( A  \  { B } ) ) )
26 uncom 3451 . . . 4  |-  ( { B }  u.  ( A  \  { B }
) )  =  ( ( A  \  { B } )  u.  { B } )
27 difsnid 3904 . . . 4  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
2826, 27syl5eq 2448 . . 3  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  =  A )
2928adantl 453 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { B }  u.  ( A  \  { B } ) )  =  A )
30 phplem1 7245 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } ) )
3125, 29, 303brtr3d 4201 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172   Ord word 4540   suc csuc 4543   omcom 4804   -1-1-onto->wf1o 5412    ~~ cen 7065
This theorem is referenced by:  phplem3  7247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-en 7069
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