Theorem sstr2 3186

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 3173	. . . 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 1890	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		dfss2 3168	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		dfss2 3168	. 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 1362   e. wcel 2164   C_ wss 3153

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  sstr  3187  sstri  3188  sseq1  3202  sseq2  3203  ssun3  3324  ssun4  3325  ssinss1  3388  ssdisj  3503  triun  4140  trintssm  4143  sspwb  4245  exss  4256  relss  4746  funss  5273  funimass2  5332  fss  5415  fiintim  6985  sbthlem2  7017  sbthlemi3  7018  sbthlemi6  7021  lsslss  13877  lspss  13895  tgss  14231  tgcl  14232  tgss3  14246  clsss  14286  neiss  14318  ssnei2  14325  cnpnei  14387  cnptopco  14390  cnptoprest  14407  txcnp  14439  neibl  14659  metcnp3  14679  bj-nntrans  15443