Theorem sstrd 3073

Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)

Hypotheses

Ref	Expression
sstrd.1	|- ( ph -> A C_ B )
sstrd.2	|- ( ph -> B C_ C )

Assertion

Ref	Expression
sstrd	|- ( ph -> A C_ C )

Proof of Theorem sstrd

Step	Hyp	Ref	Expression
1		sstrd.1	. 2 |- ( ph -> A C_ B )
2		sstrd.2	. 2 |- ( ph -> B C_ C )
3		sstr 3071	. 2 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )
4	1, 2, 3	syl2anc 406	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   C_ wss 3037

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3043  df-ss 3050

This theorem is referenced by:  sstrid  3074  syl6ss  3075  ssdif2d  3181  tfisi  4461  funss  5100  fssxp  5248  fvmptssdm  5459  suppssfv  5932  suppssov1  5933  tposss  6097  tfrlem1  6159  tfrlemibfn  6179  tfr1onlembfn  6195  tfr1onlemubacc  6197  tfr1onlemres  6200  tfrcllembfn  6208  tfrcllemubacc  6210  tfrcllemres  6213  ecinxp  6458  undifdc  6765  sbthlem1  6797  iseqf1olemnab  10154  isumss  11052  ennnfoneleminc  11769  strsetsid  11835  strleund  11890  ntrss  12131  neiint  12157  neiss  12162  restopnb  12193  iscnp4  12229  blssps  12416  blss  12417  xmettx  12499  tgqioo  12533  rescncf  12554  dvbss  12609  dvbsssg  12610  dvfgg  12612  dvcnp2cntop  12618  dvcn  12619  dvaddxxbr  12620  dvmulxxbr  12621