Theorem ssel 2939

Description: Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ssel	|- ( A C_ B -> ( C e. A -> C e. B ) )

Proof of Theorem ssel

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 2934	. . . . . 6 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
2	1	biimpi 113	. . . . 5 |- ( A C_ B -> A. x ( x e. A -> x e. B ) )
3	2	19.21bi 1450	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
4	3	anim2d 320	. . 3 |- ( A C_ B -> ( ( x = C /\ x e. A ) -> ( x = C /\ x e. B ) ) )
5	4	eximdv 1760	. 2 |- ( A C_ B -> ( E. x ( x = C /\ x e. A ) -> E. x ( x = C /\ x e. B ) ) )
6		df-clel 2036	. 2 |- ( C e. A <-> E. x ( x = C /\ x e. A ) )
7		df-clel 2036	. 2 |- ( C e. B <-> E. x ( x = C /\ x e. B ) )
8	5, 6, 7	3imtr4g 194	1 |- ( A C_ B -> ( C e. A -> C e. B ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  ssel2  2940  sseli  2941  sseld  2944  sstr2  2952  ssralv  3004  ssrexv  3005  ralss  3006  rexss  3007  ssconb  3076  sscon  3077  ssdif  3078  unss1  3112  ssrin  3162  difin2  3199  reuss2  3217  reupick  3221  sssnm  3525  uniss  3601  ss2iun  3672  ssiun  3699  iinss  3708  disjss2  3748  disjss1  3751  pwnss  3912  sspwb  3952  ssopab2b  4013  soss  4051  sucssel  4161  ssorduni  4213  onintonm  4243  onnmin  4292  ssnel  4293  wessep  4302  ssrel  4428  ssrel2  4430  ssrelrel  4440  xpss12  4445  cnvss  4508  dmss  4534  elreldm  4560  dmcosseq  4603  relssres  4648  iss  4654  resopab2  4655  issref  4707  ssrnres  4763  dfco2a  4821  cores  4824  funssres  4942  fununi  4967  funimaexglem  4982  dfimafn  5222  funimass4  5224  funimass3  5283  dff4im  5313  funfvima2  5391  funfvima3  5392  f1elima  5412  riotass2  5494  ssoprab2b  5562  resoprab2  5598  releldm2  5811  reldmtpos  5868  dmtpos  5871  rdgss  5970  eqreznegel  8549  negm  8550  iccsupr  8835  bdop  9995  bj-nnen2lp  10079