Theorem sstr2 3164

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 3151	. . . 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 1879	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		dfss2 3146	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		dfss2 3146	. 2 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
6	3, 4, 5	3imtr4g 205	1 |- ( A C_ B -> ( B C_ C -> A C_ C ) )

Syntax hints:   -> wi 4   A.wal 1351   e. wcel 2148   C_ wss 3131

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144

This theorem is referenced by:  sstr  3165  sstri  3166  sseq1  3180  sseq2  3181  ssun3  3302  ssun4  3303  ssinss1  3366  ssdisj  3481  triun  4116  trintssm  4119  sspwb  4218  exss  4229  relss  4715  funss  5237  funimass2  5296  fss  5379  fiintim  6930  sbthlem2  6959  sbthlemi3  6960  sbthlemi6  6963  lsslss  13473  tgss  13602  tgcl  13603  tgss3  13617  clsss  13657  neiss  13689  ssnei2  13696  cnpnei  13758  cnptopco  13761  cnptoprest  13778  txcnp  13810  neibl  14030  metcnp3  14050  bj-nntrans  14742