Theorem elssuni 3942

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 3258	. 2 |- A C_ A
2		ssuni 3936	. 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 2203   C_ wss 3211   U.cuni 3914

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  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-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-uni 3915

This theorem is referenced by:  unissel  3943  ssunieq  3947  pwuni  4305  pwel  4334  uniopel  4373  iunpw  4601  dmrnssfld  5020  iotaexab  5331  fvssunirng  5685  relfvssunirn  5686  sefvex  5691  riotaexg  6007  pwuninel2  6513  tfrlem9  6550  tfrexlem  6565  sbthlem1  7227  sbthlem2  7228  unirnioo  10306  eltopss  14874  toponss  14891  isbasis3g  14911  baspartn  14915  bastg  14926  tgcl  14929  epttop  14955  difopn  14973  ssntr  14987  isopn3  14990  isopn3i  15000  neiuni  15026  resttopon  15036  restopn2  15048  ssidcn  15075  lmtopcnp  15115  txuni2  15121  hmeoimaf1o  15179  tgioo  15419  bj-elssuniab  16563