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  7593  fzossnn0  10268  seq3split  10597  1arith  12561  ctinf  12672  nninfdclemcl  12690  nninfdclemp1  12692  mhmima  13193  znleval  14285  tgcl  14384  epttop  14410  ntrin  14444  cnconst2  14553  cnrest2  14556  cnptopresti  14558  cnptoprest2  14560  hmeores  14635  blin2  14752  ivthdec  14964  limcdifap  14982  limcresi  14986  dvfgg  15008  dvcnp2cntop  15019  dvaddxxbr  15021  reeff1olem  15091  domomsubct  15732