Theorem elssuni 3681

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 3044	. 2 |- A C_ A
2		ssuni 3675	. 2 |- ( ( A C_ A /\ A e. B ) -> A C_ U. B )
3	1, 2	mpan 415	1 |- ( A e. B -> A C_ U. B )

Syntax hints:   -> wi 4   e. wcel 1438   C_ wss 2999   U.cuni 3653

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654

This theorem is referenced by:  unissel  3682  ssunieq  3686  pwuni  4027  pwel  4045  uniopel  4083  iunpw  4302  dmrnssfld  4696  fvssunirng  5320  relfvssunirn  5321  sefvex  5326  riotaexg  5612  pwuninel2  6047  tfrlem9  6084  tfrexlem  6099  sbthlem1  6666  sbthlem2  6667  unirnioo  9391  eltopss  11606  toponss  11622  bj-elssuniab  11691