Theorem sstrid 3195

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 3194	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  cossxp2  5194  fimacnv  5694  smores2  6361  f1imaen2g  6861  phplem4dom  6932  isinfinf  6967  fidcenumlemrk  7029  casef  7163  genipv  7595  fzossnn0  10270  seq3split  10599  1arith  12563  ctinf  12674  nninfdclemcl  12692  nninfdclemp1  12694  mhmima  13195  znleval  14287  tgcl  14408  epttop  14434  ntrin  14468  cnconst2  14577  cnrest2  14580  cnptopresti  14582  cnptoprest2  14584  hmeores  14659  blin2  14776  ivthdec  14988  limcdifap  15006  limcresi  15010  dvfgg  15032  dvcnp2cntop  15043  dvaddxxbr  15045  reeff1olem  15115  domomsubct  15756