Theorem sstrid 3235

Description: Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)

Hypotheses

Ref	Expression
sstrid.1	|- A C_ B
sstrid.2	|- ( ph -> B C_ C )

Assertion

Ref	Expression
sstrid	|- ( ph -> A C_ C )

Proof of Theorem sstrid

Step	Hyp	Ref	Expression
1		sstrid.1	. . 3 |- A C_ B
2	1	a1i 9	. 2 |- ( ph -> A C_ B )
3		sstrid.2	. 2 |- ( ph -> B C_ C )
4	2, 3	sstrd 3234	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 3197

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  cossxp2  5252  fimass  5489  fimacnv  5766  smores2  6446  f1imaen2g  6953  phplem4dom  7031  isinfinf  7067  fidcenumlemrk  7129  casef  7263  genipv  7704  fzossnn0  10381  seq3split  10718  1arith  12898  ctinf  13009  nninfdclemcl  13027  nninfdclemp1  13029  mhmima  13532  znleval  14625  tgcl  14746  epttop  14772  ntrin  14806  cnconst2  14915  cnrest2  14918  cnptopresti  14920  cnptoprest2  14922  hmeores  14997  blin2  15114  ivthdec  15326  limcdifap  15344  limcresi  15348  dvfgg  15370  dvcnp2cntop  15381  dvaddxxbr  15383  reeff1olem  15453  domomsubct  16393