Theorem sstr2 3231

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 3218	. . . 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 1925	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		ssalel 3212	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		ssalel 3212	. 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 1393   e. wcel 2200   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  sstr  3232  sstri  3233  sseq1  3247  sseq2  3248  ssun3  3369  ssun4  3370  ssinss1  3433  ssdisj  3548  triun  4195  trintssm  4198  sspwb  4302  exss  4313  relss  4806  funss  5337  funimass2  5399  fss  5485  fiintim  7104  sbthlem2  7136  sbthlemi3  7137  sbthlemi6  7140  lsslss  14360  lspss  14378  tgss  14752  tgcl  14753  tgss3  14767  clsss  14807  neiss  14839  ssnei2  14846  cnpnei  14908  cnptopco  14911  cnptoprest  14928  txcnp  14960  neibl  15180  metcnp3  15200  bj-nntrans  16369