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Theorem xpen 6992
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 6836 . . . . 5  |-  Rel  ~~
21brrelexi 4717 . . . 4  |-  ( C 
~~  D  ->  C  e.  _V )
3 endom 6856 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
4 xpdom1g 6927 . . . 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 4718 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
7 endom 6856 . . . 4  |-  ( C 
~~  D  ->  C  ~<_  D )
8 xpdom2g 6926 . . . 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 6882 . . 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 4718 . . . 4  |-  ( C 
~~  D  ->  D  e.  _V )
13 ensym 6878 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  A )
14 endom 6856 . . . . 5  |-  ( B 
~~  A  ->  B  ~<_  A )
1513, 14syl 17 . . . 4  |-  ( A 
~~  B  ->  B  ~<_  A )
16 xpdom1g 6927 . . . 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 4717 . . . 4  |-  ( A 
~~  B  ->  A  e.  _V )
19 ensym 6878 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  C )
20 endom 6856 . . . . 5  |-  ( D 
~~  C  ->  D  ~<_  C )
2119, 20syl 17 . . . 4  |-  ( C 
~~  D  ->  D  ~<_  C )
22 xpdom2g 6926 . . . 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 6882 . . 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 6949 . 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 2763   class class class wbr 3997    X. cxp 4659    ~~ cen 6828    ~<_ cdom 6829
This theorem is referenced by:  map2xp  6999  unxpdom2  7039  sucxpdom  7040  xpnum  7552  infxpenlem  7609  infxpidm2  7612  xpcdaen  7777  mapcdaen  7778  pwcdaen  7779  cdaxpdom  7783  ackbij1lem5  7818  canthp1lem1  8242  xpnnen  12449  xpomenOLD  12451  qnnen  12454  rexpen  12468  met2ndci  18030  re2ndc  18269  dyadmbl  18917  opnmblALT  18920  mbfimaopnlem  18972  xpengOLD  25742
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-1st 6056  df-2nd 6057  df-er 6628  df-en 6832  df-dom 6833
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