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Theorem xpdom1 6781
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
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 6779 . 2  |-  ( ( C  e.  _V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
31, 2mpan 421 1  |-  ( A  ~<_  B  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2128   _Vcvv 2712   class class class wbr 3966    X. cxp 4585    ~<_ cdom 6685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-1st 6089  df-2nd 6090  df-en 6687  df-dom 6688
This theorem is referenced by: (None)
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