Theorem sstr2 3255

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 3242	. . . 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 1932	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		ssalel 3235	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		ssalel 3235	. 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 1400   e. wcel 2209   C_ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  sstr  3256  sstri  3257  sseq1  3271  sseq2  3272  ssun3  3394  ssun4  3395  ssinss1  3460  ssdisj  3580  sspw  3698  triun  4237  trintssm  4240  sspwb  4351  exss  4362  relss  4857  funss  5391  funimass2  5454  fss  5541  fiintim  7228  sbthlem2  7265  sbthlemi3  7266  sbthlemi6  7269  lsslss  14690  lspss  14708  tgss  15087  tgcl  15088  tgss3  15102  clsss  15142  neiss  15174  ssnei2  15181  cnpnei  15243  cnptopco  15246  cnptoprest  15263  txcnp  15295  neibl  15515  metcnp3  15535  bj-nntrans  16891