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Theorem phplem2 7129
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 4297 . . . . . 6  |-  { <. B ,  A >. }  e.  _V
2 phplem2.2 . . . . . . 7  |-  B  e. 
_V
3 phplem2.1 . . . . . . 7  |-  A  e. 
_V
42, 3f1osn 5596 . . . . . 6  |-  { <. B ,  A >. } : { B } -1-1-onto-> { A }
5 f1oen3g 6965 . . . . . 6  |-  ( ( { <. B ,  A >. }  e.  _V  /\  {
<. B ,  A >. } : { B } -1-1-onto-> { A } )  ->  { B }  ~~  { A }
)
61, 4, 5mp2an 653 . . . . 5  |-  { B }  ~~  { A }
7 difss 3379 . . . . . . 7  |-  ( A 
\  { B }
)  C_  A
83, 7ssexi 4240 . . . . . 6  |-  ( A 
\  { B }
)  e.  _V
98enref 6982 . . . . 5  |-  ( A 
\  { B }
)  ~~  ( A  \  { B } )
106, 9pm3.2i 441 . . . 4  |-  ( { B }  ~~  { A }  /\  ( A  \  { B }
)  ~~  ( A  \  { B } ) )
11 incom 3437 . . . . . 6  |-  ( { A }  i^i  ( A  \  { B }
) )  =  ( ( A  \  { B } )  i^i  { A } )
12 ssrin 3470 . . . . . . . . 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 4746 . . . . . . . . 9  |-  ( A  e.  om  ->  Ord  A )
15 orddisj 4512 . . . . . . . . 9  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
1614, 15syl 15 . . . . . . . 8  |-  ( A  e.  om  ->  ( A  i^i  { A }
)  =  (/) )
1713, 16syl5sseq 3302 . . . . . . 7  |-  ( A  e.  om  ->  (
( A  \  { B } )  i^i  { A } )  C_  (/) )
18 ss0 3561 . . . . . . 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 2402 . . . . 5  |-  ( A  e.  om  ->  ( { A }  i^i  ( A  \  { B }
) )  =  (/) )
21 disjdif 3602 . . . . 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 7031 . . . 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 3395 . . . 4  |-  ( { B }  u.  ( A  \  { B }
) )  =  ( ( A  \  { B } )  u.  { B } )
27 difsnid 3840 . . . 4  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
2826, 27syl5eq 2402 . . 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 7128 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } ) )
3125, 29, 303brtr3d 4133 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  ~~  ( suc 
A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    \ cdif 3225    u. cun 3226    i^i cin 3227    C_ wss 3228   (/)c0 3531   {csn 3716   <.cop 3719   class class class wbr 4104   Ord word 4473   suc csuc 4476   omcom 4738   -1-1-onto->wf1o 5336    ~~ cen 6948
This theorem is referenced by:  phplem3  7130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-en 6952
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