Theorem elssuni 3958

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 3268	. 2 |- A C_ A
2		ssuni 3952	. 2 |- ( ( A C_ A /\ A e. B ) -> A C_ U. B )
3	1, 2	mpan 428	1 |- ( A e. B -> A C_ U. B )

Syntax hints:   -> wi 4   e. wcel 2209   C_ wss 3220   U.cuni 3930

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-uni 3931

This theorem is referenced by:  unissel  3959  ssunieq  3963  pwuni  4324  pwel  4353  uniopel  4392  iunpw  4621  dmrnssfld  5040  iotaexab  5351  fvssunirng  5705  relfvssunirn  5706  sefvex  5711  riotaexg  6032  pwuninel2  6543  tfrlem9  6580  tfrexlem  6595  sbthlem1  7264  sbthlem2  7265  unirnioo  10354  eltopss  15033  toponss  15050  isbasis3g  15070  baspartn  15074  bastg  15085  tgcl  15088  epttop  15114  difopn  15132  ssntr  15146  isopn3  15149  isopn3i  15159  neiuni  15185  resttopon  15195  restopn2  15207  ssidcn  15234  lmtopcnp  15274  txuni2  15280  hmeoimaf1o  15338  tgioo  15578  bj-elssuniab  16733