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Theorem xpen 7019
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 6863 . . . . 5  |-  Rel  ~~
21brrelexi 4728 . . . 4  |-  ( C 
~~  D  ->  C  e.  _V )
3 endom 6883 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
4 xpdom1g 6954 . . . 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 4729 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
7 endom 6883 . . . 4  |-  ( C 
~~  D  ->  C  ~<_  D )
8 xpdom2g 6953 . . . 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 6909 . . 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 4729 . . . 4  |-  ( C 
~~  D  ->  D  e.  _V )
13 ensym 6905 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  A )
14 endom 6883 . . . . 5  |-  ( B 
~~  A  ->  B  ~<_  A )
1513, 14syl 17 . . . 4  |-  ( A 
~~  B  ->  B  ~<_  A )
16 xpdom1g 6954 . . . 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 4728 . . . 4  |-  ( A 
~~  B  ->  A  e.  _V )
19 ensym 6905 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  C )
20 endom 6883 . . . . 5  |-  ( D 
~~  C  ->  D  ~<_  C )
2119, 20syl 17 . . . 4  |-  ( C 
~~  D  ->  D  ~<_  C )
22 xpdom2g 6953 . . . 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 6909 . . 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 6976 . 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 1688   _Vcvv 2789   class class class wbr 4024    X. cxp 4686    ~~ cen 6855    ~<_ cdom 6856
This theorem is referenced by:  map2xp  7026  unxpdom2  7066  sucxpdom  7067  xpnum  7579  infxpenlem  7636  infxpidm2  7639  xpcdaen  7804  mapcdaen  7805  pwcdaen  7806  cdaxpdom  7810  ackbij1lem5  7845  canthp1lem1  8269  xpnnen  12481  xpomenOLD  12483  qnnen  12486  rexpen  12500  met2ndci  18062  re2ndc  18301  dyadmbl  18949  opnmblALT  18952  mbfimaopnlem  19004  xpengOLD  25774
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-1st 6083  df-2nd 6084  df-er 6655  df-en 6859  df-dom 6860
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