Theorem sstr2 3154

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 3141	. . . 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 1872	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		dfss2 3136	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		dfss2 3136	. 2 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
6	3, 4, 5	3imtr4g 204	1 |- ( A C_ B -> ( B C_ C -> A C_ C ) )

Syntax hints:   -> wi 4   A.wal 1346   e. wcel 2141   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  sstr  3155  sstri  3156  sseq1  3170  sseq2  3171  ssun3  3292  ssun4  3293  ssinss1  3356  ssdisj  3471  triun  4100  trintssm  4103  sspwb  4201  exss  4212  relss  4698  funss  5217  funimass2  5276  fss  5359  fiintim  6906  sbthlem2  6935  sbthlemi3  6936  sbthlemi6  6939  tgss  12857  tgcl  12858  tgss3  12872  clsss  12912  neiss  12944  ssnei2  12951  cnpnei  13013  cnptopco  13016  cnptoprest  13033  txcnp  13065  neibl  13285  metcnp3  13305  bj-nntrans  13986