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Theorem xpen 7261
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 7105 . . . . 5  |-  Rel  ~~
21brrelexi 4909 . . . 4  |-  ( C 
~~  D  ->  C  e.  _V )
3 endom 7125 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
4 xpdom1g 7196 . . . 4  |-  ( ( C  e.  _V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
52, 3, 4syl2anr 465 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
61brrelex2i 4910 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
7 endom 7125 . . . 4  |-  ( C 
~~  D  ->  C  ~<_  D )
8 xpdom2g 7195 . . . 4  |-  ( ( B  e.  _V  /\  C  ~<_  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
96, 7, 8syl2an 464 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
10 domtr 7151 . . 3  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  C )  /\  ( B  X.  C )  ~<_  ( B  X.  D ) )  ->  ( A  X.  C )  ~<_  ( B  X.  D ) )
115, 9, 10syl2anc 643 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  D ) )
121brrelex2i 4910 . . . 4  |-  ( C 
~~  D  ->  D  e.  _V )
13 ensym 7147 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  A )
14 endom 7125 . . . . 5  |-  ( B 
~~  A  ->  B  ~<_  A )
1513, 14syl 16 . . . 4  |-  ( A 
~~  B  ->  B  ~<_  A )
16 xpdom1g 7196 . . . 4  |-  ( ( D  e.  _V  /\  B  ~<_  A )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
1712, 15, 16syl2anr 465 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
181brrelexi 4909 . . . 4  |-  ( A 
~~  B  ->  A  e.  _V )
19 ensym 7147 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  C )
20 endom 7125 . . . . 5  |-  ( D 
~~  C  ->  D  ~<_  C )
2119, 20syl 16 . . . 4  |-  ( C 
~~  D  ->  D  ~<_  C )
22 xpdom2g 7195 . . . 4  |-  ( ( A  e.  _V  /\  D  ~<_  C )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
2318, 21, 22syl2an 464 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
24 domtr 7151 . . 3  |-  ( ( ( B  X.  D
)  ~<_  ( A  X.  D )  /\  ( A  X.  D )  ~<_  ( A  X.  C ) )  ->  ( B  X.  D )  ~<_  ( A  X.  C ) )
2517, 23, 24syl2anc 643 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  C ) )
26 sbth 7218 . 2  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  D )  /\  ( B  X.  D )  ~<_  ( A  X.  C ) )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
2711, 25, 26syl2anc 643 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   _Vcvv 2948   class class class wbr 4204    X. cxp 4867    ~~ cen 7097    ~<_ cdom 7098
This theorem is referenced by:  map2xp  7268  unxpdom2  7308  sucxpdom  7309  xpnum  7827  infxpenlem  7884  infxpidm2  7887  xpcdaen  8052  mapcdaen  8053  pwcdaen  8054  cdaxpdom  8058  ackbij1lem5  8093  canthp1lem1  8516  xpnnen  12796  xpomenOLD  12798  qnnen  12801  rexpen  12815  met2ndci  18540  re2ndc  18820  dyadmbl  19480  opnmblALT  19483  mbfimaopnlem  19535  mblfinlem  26190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-1st 6340  df-2nd 6341  df-er 6896  df-en 7101  df-dom 7102
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