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Theorem phplem2 7290
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 4408 . . . . . 6  |-  { <. B ,  A >. }  e.  _V
2 phplem2.2 . . . . . . 7  |-  B  e. 
_V
3 phplem2.1 . . . . . . 7  |-  A  e. 
_V
42, 3f1osn 5718 . . . . . 6  |-  { <. B ,  A >. } : { B } -1-1-onto-> { A }
5 f1oen3g 7126 . . . . . 6  |-  ( ( { <. B ,  A >. }  e.  _V  /\  {
<. B ,  A >. } : { B } -1-1-onto-> { A } )  ->  { B }  ~~  { A }
)
61, 4, 5mp2an 655 . . . . 5  |-  { B }  ~~  { A }
7 difss 3476 . . . . . . 7  |-  ( A 
\  { B }
)  C_  A
83, 7ssexi 4351 . . . . . 6  |-  ( A 
\  { B }
)  e.  _V
98enref 7143 . . . . 5  |-  ( A 
\  { B }
)  ~~  ( A  \  { B } )
106, 9pm3.2i 443 . . . 4  |-  ( { B }  ~~  { A }  /\  ( A  \  { B }
)  ~~  ( A  \  { B } ) )
11 incom 3535 . . . . . 6  |-  ( { A }  i^i  ( A  \  { B }
) )  =  ( ( A  \  { B } )  i^i  { A } )
12 ssrin 3568 . . . . . . . . 9  |-  ( ( A  \  { B } )  C_  A  ->  ( ( A  \  { B } )  i^i 
{ A } ) 
C_  ( A  i^i  { A } ) )
137, 12ax-mp 5 . . . . . . . 8  |-  ( ( A  \  { B } )  i^i  { A } )  C_  ( A  i^i  { A }
)
14 nnord 4856 . . . . . . . . 9  |-  ( A  e.  om  ->  Ord  A )
15 orddisj 4622 . . . . . . . . 9  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
1614, 15syl 16 . . . . . . . 8  |-  ( A  e.  om  ->  ( A  i^i  { A }
)  =  (/) )
1713, 16syl5sseq 3398 . . . . . . 7  |-  ( A  e.  om  ->  (
( A  \  { B } )  i^i  { A } )  C_  (/) )
18 ss0 3660 . . . . . . 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 2482 . . . . 5  |-  ( A  e.  om  ->  ( { A }  i^i  ( A  \  { B }
) )  =  (/) )
21 disjdif 3702 . . . . 5  |-  ( { B }  i^i  ( A  \  { B }
) )  =  (/)
2220, 21jctil 525 . . . 4  |-  ( A  e.  om  ->  (
( { B }  i^i  ( A  \  { B } ) )  =  (/)  /\  ( { A }  i^i  ( A  \  { B } ) )  =  (/) ) )
23 unen 7192 . . . 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 646 . . 3  |-  ( A  e.  om  ->  ( { B }  u.  ( A  \  { B }
) )  ~~  ( { A }  u.  ( A  \  { B }
) ) )
2524adantr 453 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { B }  u.  ( A  \  { B } ) )  ~~  ( { A }  u.  ( A  \  { B } ) ) )
26 uncom 3493 . . . 4  |-  ( { B }  u.  ( A  \  { B }
) )  =  ( ( A  \  { B } )  u.  { B } )
27 difsnid 3946 . . . 4  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
2826, 27syl5eq 2482 . . 3  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  =  A )
2928adantl 454 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { B }  u.  ( A  \  { B } ) )  =  A )
30 phplem1 7289 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } ) )
3125, 29, 303brtr3d 4244 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   class class class wbr 4215   Ord word 4583   suc csuc 4586   omcom 4848   -1-1-onto->wf1o 5456    ~~ cen 7109
This theorem is referenced by:  phplem3  7291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-en 7113
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