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Theorem php2 7496
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 2503 . . . . 5  |-  ( x  =  A  ->  (
x  e.  om  <->  A  e.  om ) )
2 psseq2 3444 . . . . 5  |-  ( x  =  A  ->  ( B  C.  x  <->  B  C.  A
) )
31, 2anbi12d 710 . . . 4  |-  ( x  =  A  ->  (
( x  e.  om  /\  B  C.  x )  <->  ( A  e.  om  /\  B  C.  A ) ) )
4 breq2 4296 . . . 4  |-  ( x  =  A  ->  ( B  ~<  x  <->  B  ~<  A ) )
53, 4imbi12d 320 . . 3  |-  ( x  =  A  ->  (
( ( x  e. 
om  /\  B  C.  x
)  ->  B  ~<  x )  <->  ( ( A  e.  om  /\  B  C.  A )  ->  B  ~<  A ) ) )
6 vex 2975 . . . . . 6  |-  x  e. 
_V
7 pssss 3451 . . . . . 6  |-  ( B 
C.  x  ->  B  C_  x )
8 ssdomg 7355 . . . . . 6  |-  ( x  e.  _V  ->  ( B  C_  x  ->  B  ~<_  x ) )
96, 7, 8mpsyl 63 . . . . 5  |-  ( B 
C.  x  ->  B  ~<_  x )
109adantl 466 . . . 4  |-  ( ( x  e.  om  /\  B  C.  x )  ->  B  ~<_  x )
11 php 7495 . . . . 5  |-  ( ( x  e.  om  /\  B  C.  x )  ->  -.  x  ~~  B )
12 ensym 7358 . . . . 5  |-  ( B 
~~  x  ->  x  ~~  B )
1311, 12nsyl 121 . . . 4  |-  ( ( x  e.  om  /\  B  C.  x )  ->  -.  B  ~~  x )
14 brsdom 7332 . . . 4  |-  ( B 
~<  x  <->  ( B  ~<_  x  /\  -.  B  ~~  x ) )
1510, 13, 14sylanbrc 664 . . 3  |-  ( ( x  e.  om  /\  B  C.  x )  ->  B  ~<  x )
165, 15vtoclg 3030 . 2  |-  ( A  e.  om  ->  (
( A  e.  om  /\  B  C.  A )  ->  B  ~<  A )
)
1716anabsi5 813 1  |-  ( ( A  e.  om  /\  B  C.  A )  ->  B  ~<  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972    C_ wss 3328    C. wpss 3329   class class class wbr 4292   omcom 6476    ~~ cen 7307    ~<_ cdom 7308    ~< csdm 7309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-om 6477  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313
This theorem is referenced by:  php4  7498  nndomo  7504
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