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  fimacnv  5764  smores2  6440  f1imaen2g  6945  phplem4dom  7023  isinfinf  7059  fidcenumlemrk  7121  casef  7255  genipv  7696  fzossnn0  10373  seq3split  10710  1arith  12890  ctinf  13001  nninfdclemcl  13019  nninfdclemp1  13021  mhmima  13524  znleval  14617  tgcl  14738  epttop  14764  ntrin  14798  cnconst2  14907  cnrest2  14910  cnptopresti  14912  cnptoprest2  14914  hmeores  14989  blin2  15106  ivthdec  15318  limcdifap  15336  limcresi  15340  dvfgg  15362  dvcnp2cntop  15373  dvaddxxbr  15375  reeff1olem  15445  domomsubct  16367