Theorem sstr2 3234

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 3221	. . . 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 1927	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		ssalel 3215	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		ssalel 3215	. 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 1395   e. wcel 2202   C_ wss 3200

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  sstr  3235  sstri  3236  sseq1  3250  sseq2  3251  ssun3  3372  ssun4  3373  ssinss1  3436  ssdisj  3551  triun  4200  trintssm  4203  sspwb  4308  exss  4319  relss  4813  funss  5345  funimass2  5408  fss  5494  fiintim  7122  sbthlem2  7156  sbthlemi3  7157  sbthlemi6  7160  lsslss  14394  lspss  14412  tgss  14786  tgcl  14787  tgss3  14801  clsss  14841  neiss  14873  ssnei2  14880  cnpnei  14942  cnptopco  14945  cnptoprest  14962  txcnp  14994  neibl  15214  metcnp3  15234  bj-nntrans  16546