Theorem sstr2 3162

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 3149	. . . 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 3144	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		dfss2 3144	. 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 3129

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 3135  df-ss 3142

This theorem is referenced by:  sstr  3163  sstri  3164  sseq1  3178  sseq2  3179  ssun3  3300  ssun4  3301  ssinss1  3364  ssdisj  3479  triun  4114  trintssm  4117  sspwb  4216  exss  4227  relss  4713  funss  5235  funimass2  5294  fss  5377  fiintim  6927  sbthlem2  6956  sbthlemi3  6957  sbthlemi6  6960  tgss  13499  tgcl  13500  tgss3  13514  clsss  13554  neiss  13586  ssnei2  13593  cnpnei  13655  cnptopco  13658  cnptoprest  13675  txcnp  13707  neibl  13927  metcnp3  13947  bj-nntrans  14639