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Theorem php2 7228
Description: Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.)
Assertion
Ref Expression
php2  |-  ( ( A  e.  om  /\  B  C.  A )  ->  B  ~<  A )

Proof of Theorem php2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2447 . . . . 5  |-  ( x  =  A  ->  (
x  e.  om  <->  A  e.  om ) )
2 psseq2 3378 . . . . 5  |-  ( x  =  A  ->  ( B  C.  x  <->  B  C.  A ) )
31, 2anbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( x  e.  om  /\  B  C.  x )  <-> 
( A  e.  om  /\  B  C.  A ) ) )
4 breq2 4157 . . . 4  |-  ( x  =  A  ->  ( B  ~<  x  <->  B  ~<  A ) )
53, 4imbi12d 312 . . 3  |-  ( x  =  A  ->  (
( ( x  e. 
om  /\  B  C.  x )  ->  B  ~<  x )  <->  ( ( A  e.  om  /\  B  C.  A )  ->  B  ~<  A ) ) )
6 vex 2902 . . . . . 6  |-  x  e. 
_V
7 pssss 3385 . . . . . 6  |-  ( B 
C.  x  ->  B  C_  x )
8 ssdomg 7089 . . . . . 6  |-  ( x  e.  _V  ->  ( B  C_  x  ->  B  ~<_  x ) )
96, 7, 8mpsyl 61 . . . . 5  |-  ( B 
C.  x  ->  B  ~<_  x )
109adantl 453 . . . 4  |-  ( ( x  e.  om  /\  B  C.  x )  ->  B  ~<_  x )
11 php 7227 . . . . 5  |-  ( ( x  e.  om  /\  B  C.  x )  ->  -.  x  ~~  B )
12 ensym 7092 . . . . 5  |-  ( B 
~~  x  ->  x  ~~  B )
1311, 12nsyl 115 . . . 4  |-  ( ( x  e.  om  /\  B  C.  x )  ->  -.  B  ~~  x )
14 brsdom 7066 . . . 4  |-  ( B 
~<  x  <->  ( B  ~<_  x  /\  -.  B  ~~  x ) )
1510, 13, 14sylanbrc 646 . . 3  |-  ( ( x  e.  om  /\  B  C.  x )  ->  B  ~<  x )
165, 15vtoclg 2954 . 2  |-  ( A  e.  om  ->  (
( A  e.  om  /\  B  C.  A )  ->  B  ~<  A ) )
1716anabsi5 791 1  |-  ( ( A  e.  om  /\  B  C.  A )  ->  B  ~<  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263    C. wpss 3264   class class class wbr 4153   omcom 4785    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044
This theorem is referenced by:  php4  7230  nndomo  7236
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048
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