Theorem sstr2 3245

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 3232	. . . 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 1928	. 2 |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) )
4		ssalel 3226	. 2 |- ( B C_ C <-> A. x ( x e. B -> x e. C ) )
5		ssalel 3226	. 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 1396   e. wcel 2203   C_ wss 3211

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-ss 3224

This theorem is referenced by:  sstr  3246  sstri  3247  sseq1  3261  sseq2  3262  ssun3  3384  ssun4  3385  ssinss1  3450  ssdisj  3565  sspw  3682  triun  4221  trintssm  4224  sspwb  4332  exss  4343  relss  4837  funss  5371  funimass2  5434  fss  5521  fiintim  7191  sbthlem2  7228  sbthlemi3  7229  sbthlemi6  7232  lsslss  14529  lspss  14547  tgss  14928  tgcl  14929  tgss3  14943  clsss  14983  neiss  15015  ssnei2  15022  cnpnei  15084  cnptopco  15087  cnptoprest  15104  txcnp  15136  neibl  15356  metcnp3  15376  bj-nntrans  16721