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Theorem xpen 6909
Description: Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpen  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )

Proof of Theorem xpen
StepHypRef Expression
1 relen 6754 . . . . 5  |-  Rel  ~~
21brrelexi 4636 . . . 4  |-  ( C 
~~  D  ->  C  e.  _V )
3 endom 6774 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
4 xpdom1g 6844 . . . 4  |-  ( ( C  e.  _V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
52, 3, 4syl2anr 466 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
61brrelex2i 4637 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
7 endom 6774 . . . 4  |-  ( C 
~~  D  ->  C  ~<_  D )
8 xpdom2g 6843 . . . 4  |-  ( ( B  e.  _V  /\  C  ~<_  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
96, 7, 8syl2an 465 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
10 domtr 6799 . . 3  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  C )  /\  ( B  X.  C )  ~<_  ( B  X.  D ) )  ->  ( A  X.  C )  ~<_  ( B  X.  D ) )
115, 9, 10syl2anc 645 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  D ) )
121brrelex2i 4637 . . . 4  |-  ( C 
~~  D  ->  D  e.  _V )
13 ensym 6796 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  A )
14 endom 6774 . . . . 5  |-  ( B 
~~  A  ->  B  ~<_  A )
1513, 14syl 17 . . . 4  |-  ( A 
~~  B  ->  B  ~<_  A )
16 xpdom1g 6844 . . . 4  |-  ( ( D  e.  _V  /\  B  ~<_  A )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
1712, 15, 16syl2anr 466 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
181brrelexi 4636 . . . 4  |-  ( A 
~~  B  ->  A  e.  _V )
19 ensym 6796 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  C )
20 endom 6774 . . . . 5  |-  ( D 
~~  C  ->  D  ~<_  C )
2119, 20syl 17 . . . 4  |-  ( C 
~~  D  ->  D  ~<_  C )
22 xpdom2g 6843 . . . 4  |-  ( ( A  e.  _V  /\  D  ~<_  C )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
2318, 21, 22syl2an 465 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
24 domtr 6799 . . 3  |-  ( ( ( B  X.  D
)  ~<_  ( A  X.  D )  /\  ( A  X.  D )  ~<_  ( A  X.  C ) )  ->  ( B  X.  D )  ~<_  ( A  X.  C ) )
2517, 23, 24syl2anc 645 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  C ) )
26 sbth 6866 . 2  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  D )  /\  ( B  X.  D )  ~<_  ( A  X.  C ) )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
2711, 25, 26syl2anc 645 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   _Vcvv 2727   class class class wbr 3920    X. cxp 4578    ~~ cen 6746    ~<_ cdom 6747
This theorem is referenced by:  map2xp  6916  unxpdom2  6956  sucxpdom  6957  xpnum  7468  infxpenlem  7525  infxpidm2  7528  xpcdaen  7693  mapcdaen  7694  pwcdaen  7695  cdaxpdom  7699  ackbij1lem5  7734  canthp1lem1  8154  xpnnen  12361  xpomenOLD  12363  qnnen  12366  rexpen  12380  met2ndci  17900  re2ndc  18139  dyadmbl  18787  opnmblALT  18790  mbfimaopnlem  18842  xpengOLD  25541
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-1st 5974  df-2nd 5975  df-er 6546  df-en 6750  df-dom 6751
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