Theorem domentr 6941

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 6912	. 2 |- ( B ~~ C -> B ~<_ C )
2		domtr 6935	. 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 4082   ~~ cen 6883   ~<_ cdom 6884

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-f1o 5324  df-en 6886  df-dom 6887

This theorem is referenced by:  xpdom1g  6988  domen2  7000  phplem4dom  7019  phpm  7023  fisbth  7041  infnfi  7053  fientri3  7073  exmidfodomrlemr  7376  exmidfodomrlemrALT  7377  hashennnuni  10996  xpct  12962  umgrislfupgrenlem  15922  lfgrnloopen  15925  pwf1oexmid  16324  sbthom  16353