Theorem sstrid 3259

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 3258	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  cossxp2  5306  fimass  5545  fimacnv  5828  smores2  6555  f1imaen2g  7070  phplem4dom  7153  isinfinf  7191  fidcenumlemrk  7261  casef  7418  genipv  7866  fzossnn0  10562  seq3split  10903  1arith  13124  ballotfilemsima  13237  ctinf  13299  nninfdclemcl  13317  nninfdclemp1  13319  mhmima  13775  znleval  14960  tgcl  15088  epttop  15114  ntrin  15148  cnconst2  15257  cnrest2  15260  cnptopresti  15262  cnptoprest2  15264  hmeores  15339  blin2  15456  ivthdec  15668  limcdifap  15686  limcresi  15690  dvfgg  15712  dvcnp2cntop  15723  dvaddxxbr  15725  reeff1olem  15795  domomsubct  16945