Theorem sstr2 3149

Description: Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
sstr2	|- ( A C_ B -> ( B C_ C -> A C_ C ) )

Proof of Theorem sstr2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3136	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	imim1d 75	. . 3 |- ( A C_ B -> ( ( x e. B -> x e. C ) -> ( x e. A -> x e. C ) ) )
3	2	alimdv 1867	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		dfss2 3131	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		dfss2 3131	. 2 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
6	3, 4, 5	3imtr4g 204	1 |- ( A C_ B -> ( B C_ C -> A C_ C ) )

Syntax hints:   -> wi 4   A.wal 1341   e. wcel 2136   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  sstr  3150  sstri  3151  sseq1  3165  sseq2  3166  ssun3  3287  ssun4  3288  ssinss1  3351  ssdisj  3465  triun  4093  trintssm  4096  sspwb  4194  exss  4205  relss  4691  funss  5207  funimass2  5266  fss  5349  fiintim  6894  sbthlem2  6923  sbthlemi3  6924  sbthlemi6  6927  tgss  12703  tgcl  12704  tgss3  12718  clsss  12758  neiss  12790  ssnei2  12797  cnpnei  12859  cnptopco  12862  cnptoprest  12879  txcnp  12911  neibl  13131  metcnp3  13151  bj-nntrans  13833