Theorem sstr2 3187

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 3174	. . . 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 1890	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		dfss2 3169	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		dfss2 3169	. 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 1362   e. wcel 2164   C_ wss 3154

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3160  df-ss 3167

This theorem is referenced by:  sstr  3188  sstri  3189  sseq1  3203  sseq2  3204  ssun3  3325  ssun4  3326  ssinss1  3389  ssdisj  3504  triun  4141  trintssm  4144  sspwb  4246  exss  4257  relss  4747  funss  5274  funimass2  5333  fss  5416  fiintim  6987  sbthlem2  7019  sbthlemi3  7020  sbthlemi6  7023  lsslss  13880  lspss  13898  tgss  14242  tgcl  14243  tgss3  14257  clsss  14297  neiss  14329  ssnei2  14336  cnpnei  14398  cnptopco  14401  cnptoprest  14418  txcnp  14450  neibl  14670  metcnp3  14690  bj-nntrans  15513