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Theorem xpdom1 6894
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 6892 . 2  |-  ( ( C  e.  _V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
31, 2mpan 654 1  |-  ( A  ~<_  B  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   _Vcvv 2740   class class class wbr 3963    X. cxp 4624    ~<_ cdom 6794
This theorem is referenced by:  cdadom1  7745  uniimadom  8099  unirnfdomd  8122  alephreg  8137  inar1  8330  2ndcctbss  17108  tx1stc  17271  tx2ndc  17272  mbfimaopnlem  18937
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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-1st 6021  df-2nd 6022  df-en 6797  df-dom 6798
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