ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  endomtr Unicode version

Theorem endomtr 6650
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 6623 . 2  |-  ( A 
~~  B  ->  A  ~<_  B )
2 domtr 6645 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  C )  ->  A  ~<_  C )
31, 2sylan 279 1  |-  ( ( A  ~~  B  /\  B  ~<_  C )  ->  A  ~<_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   class class class wbr 3897    ~~ cen 6598    ~<_ cdom 6599
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-f1o 5098  df-en 6601  df-dom 6602
This theorem is referenced by:  cnvct  6669  xpdom1g  6693  xpdom3m  6694  domen1  6702  mapdom1g  6707  phplem4dom  6722  phpm  6725  fict  6728  fisbth  6743  fientri3  6769  difinfsn  6951  qnnen  11839  pw1dom2  12992
  Copyright terms: Public domain W3C validator