MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phplem2 Unicode version

Theorem phplem2 7041
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 4216 . . . . . 6  |-  { <. B ,  A >. }  e.  _V
2 phplem2.2 . . . . . . 7  |-  B  e. 
_V
3 phplem2.1 . . . . . . 7  |-  A  e. 
_V
42, 3f1osn 5513 . . . . . 6  |-  { <. B ,  A >. } : { B } -1-1-onto-> { A }
5 f1oen3g 6877 . . . . . 6  |-  ( ( { <. B ,  A >. }  e.  _V  /\  {
<. B ,  A >. } : { B } -1-1-onto-> { A } )  ->  { B }  ~~  { A }
)
61, 4, 5mp2an 653 . . . . 5  |-  { B }  ~~  { A }
7 difss 3303 . . . . . . 7  |-  ( A 
\  { B }
)  C_  A
83, 7ssexi 4159 . . . . . 6  |-  ( A 
\  { B }
)  e.  _V
98enref 6894 . . . . 5  |-  ( A 
\  { B }
)  ~~  ( A  \  { B } )
106, 9pm3.2i 441 . . . 4  |-  ( { B }  ~~  { A }  /\  ( A  \  { B }
)  ~~  ( A  \  { B } ) )
11 incom 3361 . . . . . 6  |-  ( { A }  i^i  ( A  \  { B }
) )  =  ( ( A  \  { B } )  i^i  { A } )
12 ssrin 3394 . . . . . . . . 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 4664 . . . . . . . . 9  |-  ( A  e.  om  ->  Ord  A )
15 orddisj 4430 . . . . . . . . 9  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
1614, 15syl 15 . . . . . . . 8  |-  ( A  e.  om  ->  ( A  i^i  { A }
)  =  (/) )
1713, 16syl5sseq 3226 . . . . . . 7  |-  ( A  e.  om  ->  (
( A  \  { B } )  i^i  { A } )  C_  (/) )
18 ss0 3485 . . . . . . 7  |-  ( ( ( A  \  { B } )  i^i  { A } )  C_  (/)  ->  (
( A  \  { B } )  i^i  { A } )  =  (/) )
1917, 18syl 15 . . . . . 6  |-  ( A  e.  om  ->  (
( A  \  { B } )  i^i  { A } )  =  (/) )
2011, 19syl5eq 2327 . . . . 5  |-  ( A  e.  om  ->  ( { A }  i^i  ( A  \  { B }
) )  =  (/) )
21 disjdif 3526 . . . . 5  |-  ( { B }  i^i  ( A  \  { B }
) )  =  (/)
2220, 21jctil 523 . . . 4  |-  ( A  e.  om  ->  (
( { B }  i^i  ( A  \  { B } ) )  =  (/)  /\  ( { A }  i^i  ( A  \  { B } ) )  =  (/) ) )
23 unen 6943 . . . 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 644 . . 3  |-  ( A  e.  om  ->  ( { B }  u.  ( A  \  { B }
) )  ~~  ( { A }  u.  ( A  \  { B }
) ) )
2524adantr 451 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { B }  u.  ( A  \  { B } ) )  ~~  ( { A }  u.  ( A  \  { B } ) ) )
26 uncom 3319 . . . 4  |-  ( { B }  u.  ( A  \  { B }
) )  =  ( ( A  \  { B } )  u.  { B } )
27 difsnid 3761 . . . 4  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
2826, 27syl5eq 2327 . . 3  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  =  A )
2928adantl 452 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { B }  u.  ( A  \  { B } ) )  =  A )
30 phplem1 7040 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } ) )
3125, 29, 303brtr3d 4052 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   Ord word 4391   suc csuc 4394   omcom 4656   -1-1-onto->wf1o 5254    ~~ cen 6860
This theorem is referenced by:  phplem3  7042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-en 6864
  Copyright terms: Public domain W3C validator