Theorem sstrid 3204

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 3203	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  cossxp2  5206  fimacnv  5709  smores2  6380  f1imaen2g  6885  phplem4dom  6959  isinfinf  6994  fidcenumlemrk  7056  casef  7190  genipv  7622  fzossnn0  10299  seq3split  10633  1arith  12690  ctinf  12801  nninfdclemcl  12819  nninfdclemp1  12821  mhmima  13323  znleval  14415  tgcl  14536  epttop  14562  ntrin  14596  cnconst2  14705  cnrest2  14708  cnptopresti  14710  cnptoprest2  14712  hmeores  14787  blin2  14904  ivthdec  15116  limcdifap  15134  limcresi  15138  dvfgg  15160  dvcnp2cntop  15171  dvaddxxbr  15173  reeff1olem  15243  domomsubct  15938