Theorem sstr2 3104

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 3091	. . . 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 1851	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		dfss2 3086	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		dfss2 3086	. 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 1329   e. wcel 1480   C_ wss 3071

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  sstr  3105  sstri  3106  sseq1  3120  sseq2  3121  ssun3  3241  ssun4  3242  ssinss1  3305  ssdisj  3419  triun  4039  trintssm  4042  sspwb  4138  exss  4149  relss  4626  funss  5142  funimass2  5201  fss  5284  fiintim  6817  sbthlem2  6846  sbthlemi3  6847  sbthlemi6  6850  tgss  12232  tgcl  12233  tgss3  12247  clsss  12287  neiss  12319  ssnei2  12326  cnpnei  12388  cnptopco  12391  cnptoprest  12408  txcnp  12440  neibl  12660  metcnp3  12680  bj-nntrans  13149