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Theorem phplem2 7690
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 4678 . . . . . 6  |-  { <. B ,  A >. }  e.  _V
2 phplem2.2 . . . . . . 7  |-  B  e. 
_V
3 phplem2.1 . . . . . . 7  |-  A  e. 
_V
42, 3f1osn 5835 . . . . . 6  |-  { <. B ,  A >. } : { B } -1-1-onto-> { A }
5 f1oen3g 7524 . . . . . 6  |-  ( ( { <. B ,  A >. }  e.  _V  /\  {
<. B ,  A >. } : { B } -1-1-onto-> { A } )  ->  { B }  ~~  { A }
)
61, 4, 5mp2an 670 . . . . 5  |-  { B }  ~~  { A }
7 difss 3617 . . . . . . 7  |-  ( A 
\  { B }
)  C_  A
83, 7ssexi 4582 . . . . . 6  |-  ( A 
\  { B }
)  e.  _V
98enref 7541 . . . . 5  |-  ( A 
\  { B }
)  ~~  ( A  \  { B } )
106, 9pm3.2i 453 . . . 4  |-  ( { B }  ~~  { A }  /\  ( A  \  { B }
)  ~~  ( A  \  { B } ) )
11 incom 3677 . . . . . 6  |-  ( { A }  i^i  ( A  \  { B }
) )  =  ( ( A  \  { B } )  i^i  { A } )
12 ssrin 3709 . . . . . . . . 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 6681 . . . . . . . . 9  |-  ( A  e.  om  ->  Ord  A )
15 orddisj 4905 . . . . . . . . 9  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
1614, 15syl 16 . . . . . . . 8  |-  ( A  e.  om  ->  ( A  i^i  { A }
)  =  (/) )
1713, 16syl5sseq 3537 . . . . . . 7  |-  ( A  e.  om  ->  (
( A  \  { B } )  i^i  { A } )  C_  (/) )
18 ss0 3815 . . . . . . 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 2507 . . . . 5  |-  ( A  e.  om  ->  ( { A }  i^i  ( A  \  { B }
) )  =  (/) )
21 disjdif 3888 . . . . 5  |-  ( { B }  i^i  ( A  \  { B }
) )  =  (/)
2220, 21jctil 535 . . . 4  |-  ( A  e.  om  ->  (
( { B }  i^i  ( A  \  { B } ) )  =  (/)  /\  ( { A }  i^i  ( A  \  { B } ) )  =  (/) ) )
23 unen 7591 . . . 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 661 . . 3  |-  ( A  e.  om  ->  ( { B }  u.  ( A  \  { B }
) )  ~~  ( { A }  u.  ( A  \  { B }
) ) )
2524adantr 463 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { B }  u.  ( A  \  { B } ) )  ~~  ( { A }  u.  ( A  \  { B } ) ) )
26 uncom 3634 . . . 4  |-  ( { B }  u.  ( A  \  { B }
) )  =  ( ( A  \  { B } )  u.  { B } )
27 difsnid 4162 . . . 4  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
2826, 27syl5eq 2507 . . 3  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  =  A )
2928adantl 464 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { B }  u.  ( A  \  { B } ) )  =  A )
30 phplem1 7689 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } ) )
3125, 29, 303brtr3d 4468 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   <.cop 4022   class class class wbr 4439   Ord word 4866   suc csuc 4869   -1-1-onto->wf1o 5569   omcom 6673    ~~ cen 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-om 6674  df-en 7510
This theorem is referenced by:  phplem3  7691
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