Theorem domentr 6960

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

Step	Hyp	Ref	Expression
1		endom 6931	. 2 |- ( B ~~ C -> B ~<_ C )
2		domtr 6954	. 2 |- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
3	1, 2	sylan2 286	1 |- ( ( A ~<_ B /\ B ~~ C ) -> A ~<_ C )

Syntax hints:   -> wi 4   /\ wa 104   class class class wbr 4086   ~~ cen 6902   ~<_ cdom 6903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-f1o 5331  df-en 6905  df-dom 6906

This theorem is referenced by:  xpdom1g  7012  domen2  7024  phplem4dom  7043  phpm  7047  fisbth  7065  infnfi  7077  fientri3  7100  exmidfodomrlemr  7403  exmidfodomrlemrALT  7404  hashennnuni  11031  xpct  13007  umgrislfupgrenlem  15969  lfgrnloopen  15972  pwf1oexmid  16536  sbthom  16566