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 ) )

This proof depends on syntax axioms:   -> wi 4   A.wal 1400   e. wcel 2209   C_ wss 3220

This proof depends on 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 proof 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 used by:  sstr  3256  sstri  3257  sseq1  3271  sseq2  3272  ssun3  3394  ssun4  3395  ssinss1  3460  ssdisj  3581  sspw  3702  triun  4244  trintssm  4247  sspwb  4358  exss  4369  relss  4864  funss  5399  funimass2  5462  fss  5549  fiintim  7242  sbthlem2  7280  sbthlemi3  7281  sbthlemi6  7284  lsslss  14827  lspss  14845  aspss  15128  tgss  15289  tgcl  15290  tgss3  15304  clsss  15344  neiss  15376  ssnei2  15383  cnpnei  15445  cnptopco  15448  cnptoprest  15465  txcnp  15497  neibl  15717  metcnp3  15737  bj-nntrans  17182