Theorem elssuni 3878

Description: An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
elssuni	|- ( A e. B -> A C_ U. B )

Proof of Theorem elssuni

Step	Hyp	Ref	Expression
1		ssid 3213	. 2 |- A C_ A
2		ssuni 3872	. 2 |- ( ( A C_ A /\ A e. B ) -> A C_ U. B )
3	1, 2	mpan 424	1 |- ( A e. B -> A C_ U. B )

Syntax hints:   -> wi 4   e. wcel 2176   C_ wss 3166   U.cuni 3850

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851

This theorem is referenced by:  unissel  3879  ssunieq  3883  pwuni  4236  pwel  4262  uniopel  4301  iunpw  4527  dmrnssfld  4941  iotaexab  5250  fvssunirng  5591  relfvssunirn  5592  sefvex  5597  riotaexg  5903  pwuninel2  6368  tfrlem9  6405  tfrexlem  6420  sbthlem1  7059  sbthlem2  7060  unirnioo  10095  eltopss  14481  toponss  14498  isbasis3g  14518  baspartn  14522  bastg  14533  tgcl  14536  epttop  14562  difopn  14580  ssntr  14594  isopn3  14597  isopn3i  14607  neiuni  14633  resttopon  14643  restopn2  14655  ssidcn  14682  lmtopcnp  14722  txuni2  14728  hmeoimaf1o  14786  tgioo  15026  bj-elssuniab  15727