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Theorem endomtr 6270
Description: Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
Assertion
Ref Expression
endomtr  |-  ( ( A  ~~  B  /\  B  ~<_  C )  ->  A  ~<_  C )

Proof of Theorem endomtr
StepHypRef Expression
1 endom 6243 . 2  |-  ( A 
~~  B  ->  A  ~<_  B )
2 domtr 6265 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  C )  ->  A  ~<_  C )
31, 2sylan 267 1  |-  ( ( A  ~~  B  /\  B  ~<_  C )  ->  A  ~<_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   class class class wbr 3764    ~~ cen 6219    ~<_ cdom 6220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-f1o 4909  df-en 6222  df-dom 6223
This theorem is referenced by:  xpdom1g  6307  xpdom3m  6308  domen1  6313  phplem4dom  6324  phpm  6327  fisbth  6340  fientri3  6353
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