Theorem sstrid 3152

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 3151	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3121  df-ss 3128

This theorem is referenced by:  cossxp2  5126  fimacnv  5613  smores2  6258  f1imaen2g  6755  phplem4dom  6824  isinfinf  6859  fidcenumlemrk  6915  casef  7049  genipv  7446  fzossnn0  10106  seq3split  10410  1arith  12293  ctinf  12359  nninfdclemcl  12377  nninfdclemp1  12379  tgcl  12664  epttop  12690  ntrin  12724  cnconst2  12833  cnrest2  12836  cnptopresti  12838  cnptoprest2  12840  hmeores  12915  blin2  13032  ivthdec  13222  limcdifap  13231  limcresi  13235  dvfgg  13257  dvcnp2cntop  13263  dvaddxxbr  13265  reeff1olem  13292