Theorem sstrid 3238

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 3237	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 3200

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  cossxp2  5260  fimass  5498  fimacnv  5776  smores2  6460  f1imaen2g  6967  phplem4dom  7048  isinfinf  7086  fidcenumlemrk  7153  casef  7287  genipv  7729  fzossnn0  10412  seq3split  10751  1arith  12958  ctinf  13069  nninfdclemcl  13087  nninfdclemp1  13089  mhmima  13592  znleval  14686  tgcl  14807  epttop  14833  ntrin  14867  cnconst2  14976  cnrest2  14979  cnptopresti  14981  cnptoprest2  14983  hmeores  15058  blin2  15175  ivthdec  15387  limcdifap  15405  limcresi  15409  dvfgg  15431  dvcnp2cntop  15442  dvaddxxbr  15444  reeff1olem  15514  domomsubct  16653