Theorem elssuni 3915

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 3244	. 2 |- A C_ A
2		ssuni 3909	. 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 2200   C_ wss 3197   U.cuni 3887

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888

This theorem is referenced by:  unissel  3916  ssunieq  3920  pwuni  4275  pwel  4303  uniopel  4342  iunpw  4570  dmrnssfld  4986  iotaexab  5296  fvssunirng  5641  relfvssunirn  5642  sefvex  5647  riotaexg  5957  pwuninel2  6426  tfrlem9  6463  tfrexlem  6478  sbthlem1  7120  sbthlem2  7121  unirnioo  10165  eltopss  14677  toponss  14694  isbasis3g  14714  baspartn  14718  bastg  14729  tgcl  14732  epttop  14758  difopn  14776  ssntr  14790  isopn3  14793  isopn3i  14803  neiuni  14829  resttopon  14839  restopn2  14851  ssidcn  14878  lmtopcnp  14918  txuni2  14924  hmeoimaf1o  14982  tgioo  15222  bj-elssuniab  16113