Theorem sstr2 3144

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 3131	. . . 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 1866	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		dfss2 3126	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		dfss2 3126	. 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 1340   e. wcel 2135   C_ wss 3111

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-in 3117  df-ss 3124

This theorem is referenced by:  sstr  3145  sstri  3146  sseq1  3160  sseq2  3161  ssun3  3282  ssun4  3283  ssinss1  3346  ssdisj  3460  triun  4087  trintssm  4090  sspwb  4188  exss  4199  relss  4685  funss  5201  funimass2  5260  fss  5343  fiintim  6885  sbthlem2  6914  sbthlemi3  6915  sbthlemi6  6918  tgss  12604  tgcl  12605  tgss3  12619  clsss  12659  neiss  12691  ssnei2  12698  cnpnei  12760  cnptopco  12763  cnptoprest  12780  txcnp  12812  neibl  13032  metcnp3  13052  bj-nntrans  13668