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Theorem xpen 7024
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 6868 . . . . 5  |-  Rel  ~~
21brrelexi 4729 . . . 4  |-  ( C 
~~  D  ->  C  e.  _V )
3 endom 6888 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
4 xpdom1g 6959 . . . 4  |-  ( ( C  e.  _V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
52, 3, 4syl2anr 464 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
61brrelex2i 4730 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
7 endom 6888 . . . 4  |-  ( C 
~~  D  ->  C  ~<_  D )
8 xpdom2g 6958 . . . 4  |-  ( ( B  e.  _V  /\  C  ~<_  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
96, 7, 8syl2an 463 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
10 domtr 6914 . . 3  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  C )  /\  ( B  X.  C )  ~<_  ( B  X.  D ) )  ->  ( A  X.  C )  ~<_  ( B  X.  D ) )
115, 9, 10syl2anc 642 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  D ) )
121brrelex2i 4730 . . . 4  |-  ( C 
~~  D  ->  D  e.  _V )
13 ensym 6910 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  A )
14 endom 6888 . . . . 5  |-  ( B 
~~  A  ->  B  ~<_  A )
1513, 14syl 15 . . . 4  |-  ( A 
~~  B  ->  B  ~<_  A )
16 xpdom1g 6959 . . . 4  |-  ( ( D  e.  _V  /\  B  ~<_  A )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
1712, 15, 16syl2anr 464 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
181brrelexi 4729 . . . 4  |-  ( A 
~~  B  ->  A  e.  _V )
19 ensym 6910 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  C )
20 endom 6888 . . . . 5  |-  ( D 
~~  C  ->  D  ~<_  C )
2119, 20syl 15 . . . 4  |-  ( C 
~~  D  ->  D  ~<_  C )
22 xpdom2g 6958 . . . 4  |-  ( ( A  e.  _V  /\  D  ~<_  C )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
2318, 21, 22syl2an 463 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
24 domtr 6914 . . 3  |-  ( ( ( B  X.  D
)  ~<_  ( A  X.  D )  /\  ( A  X.  D )  ~<_  ( A  X.  C ) )  ->  ( B  X.  D )  ~<_  ( A  X.  C ) )
2517, 23, 24syl2anc 642 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  C ) )
26 sbth 6981 . 2  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  D )  /\  ( B  X.  D )  ~<_  ( A  X.  C ) )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
2711, 25, 26syl2anc 642 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    X. cxp 4687    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  map2xp  7031  unxpdom2  7071  sucxpdom  7072  xpnum  7584  infxpenlem  7641  infxpidm2  7644  xpcdaen  7809  mapcdaen  7810  pwcdaen  7811  cdaxpdom  7815  ackbij1lem5  7850  canthp1lem1  8274  xpnnen  12487  xpomenOLD  12489  qnnen  12492  rexpen  12506  met2ndci  18068  re2ndc  18307  dyadmbl  18955  opnmblALT  18958  mbfimaopnlem  19010  xpengOLD  26375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-2nd 6123  df-er 6660  df-en 6864  df-dom 6865
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