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Theorem xpdom1 7198
Description: Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Mar-2006.)
Hypothesis
Ref Expression
xpdom1.2  |-  C  e. 
_V
Assertion
Ref Expression
xpdom1  |-  ( A  ~<_  B  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )

Proof of Theorem xpdom1
StepHypRef Expression
1 xpdom1.2 . 2  |-  C  e. 
_V
2 xpdom1g 7196 . 2  |-  ( ( C  e.  _V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
31, 2mpan 652 1  |-  ( A  ~<_  B  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   _Vcvv 2948   class class class wbr 4204    X. cxp 4867    ~<_ cdom 7098
This theorem is referenced by:  cdadom1  8055  uniimadom  8408  unirnfdomd  8431  alephreg  8446  inar1  8639  2ndcctbss  17506  tx1stc  17670  tx2ndc  17671  mbfimaopnlem  19535
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-en 7101  df-dom 7102
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