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  7132  casef  7266  genipv  7707  fzossnn0  10385  seq3split  10722  1arith  12906  ctinf  13017  nninfdclemcl  13035  nninfdclemp1  13037  mhmima  13540  znleval  14633  tgcl  14754  epttop  14780  ntrin  14814  cnconst2  14923  cnrest2  14926  cnptopresti  14928  cnptoprest2  14930  hmeores  15005  blin2  15122  ivthdec  15334  limcdifap  15352  limcresi  15356  dvfgg  15378  dvcnp2cntop  15389  dvaddxxbr  15391  reeff1olem  15461  domomsubct  16454