Theorem elssuni 3608

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 2964	. 2 |- A C_ A
2		ssuni 3602	. 2 |- ( ( A C_ A /\ A e. B ) -> A C_ U. B )
3	1, 2	mpan 400	1 |- ( A e. B -> A C_ U. B )

Syntax hints:   -> wi 4   e. wcel 1393   C_ wss 2917   U.cuni 3580

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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  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-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581

This theorem is referenced by:  unissel  3609  ssunieq  3613  pwuni  3943  pwel  3954  uniopel  3993  iunpw  4211  dmrnssfld  4595  fvssunirng  5190  relfvssunirn  5191  sefvex  5196  riotaexg  5472  pwuninel2  5897  tfrlem9  5935  tfrexlem  5948  unirnioo  8842  bj-elssuniab  9930