Theorem sstr2 3200

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 3187	. . . 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 1902	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		ssalel 3181	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		ssalel 3181	. 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 1371   e. wcel 2176   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  sstr  3201  sstri  3202  sseq1  3216  sseq2  3217  ssun3  3338  ssun4  3339  ssinss1  3402  ssdisj  3517  triun  4155  trintssm  4158  sspwb  4260  exss  4271  relss  4762  funss  5290  funimass2  5352  fss  5437  fiintim  7028  sbthlem2  7060  sbthlemi3  7061  sbthlemi6  7064  lsslss  14143  lspss  14161  tgss  14535  tgcl  14536  tgss3  14550  clsss  14590  neiss  14622  ssnei2  14629  cnpnei  14691  cnptopco  14694  cnptoprest  14711  txcnp  14743  neibl  14963  metcnp3  14983  bj-nntrans  15887