Theorem sstrid 3113

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 3112	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089

This theorem is referenced by:  cossxp2  5070  fimacnv  5557  smores2  6199  f1imaen2g  6695  phplem4dom  6764  isinfinf  6799  fidcenumlemrk  6850  casef  6981  genipv  7341  fzossnn0  9983  seq3split  10283  ctinf  11979  tgcl  12272  epttop  12298  ntrin  12332  cnconst2  12441  cnrest2  12444  cnptopresti  12446  cnptoprest2  12448  hmeores  12523  blin2  12640  ivthdec  12830  limcdifap  12839  limcresi  12843  dvfgg  12865  dvcnp2cntop  12871  dvaddxxbr  12873  reeff1olem  12900