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Theorem domentr 6685
Description: Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
Assertion
Ref Expression
domentr  |-  ( ( A  ~<_  B  /\  B  ~~  C )  ->  A  ~<_  C )

Proof of Theorem domentr
StepHypRef Expression
1 endom 6657 . 2  |-  ( B 
~~  C  ->  B  ~<_  C )
2 domtr 6679 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  C )  ->  A  ~<_  C )
31, 2sylan2 284 1  |-  ( ( A  ~<_  B  /\  B  ~~  C )  ->  A  ~<_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   class class class wbr 3929    ~~ cen 6632    ~<_ cdom 6633
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-f1o 5130  df-en 6635  df-dom 6636
This theorem is referenced by:  xpdom1g  6727  domen2  6737  phplem4dom  6756  phpm  6759  fisbth  6777  infnfi  6789  fientri3  6803  exmidfodomrlemr  7058  exmidfodomrlemrALT  7059  hashennnuni  10532  xpct  11916  pwf1oexmid  13224  sbthom  13251
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