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Theorem elssuni 3838
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
StepHypRef Expression
1 ssid 3176 . 2  |-  A  C_  A
2 ssuni 3832 . 2  |-  ( ( A  C_  A  /\  A  e.  B )  ->  A  C_  U. B )
31, 2mpan 424 1  |-  ( A  e.  B  ->  A  C_ 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148    C_ wss 3130   U.cuni 3810
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-uni 3811
This theorem is referenced by:  unissel  3839  ssunieq  3843  pwuni  4193  pwel  4219  uniopel  4257  iunpw  4481  dmrnssfld  4891  fvssunirng  5531  relfvssunirn  5532  sefvex  5537  riotaexg  5835  pwuninel2  6283  tfrlem9  6320  tfrexlem  6335  sbthlem1  6956  sbthlem2  6957  unirnioo  9973  eltopss  13512  toponss  13529  isbasis3g  13549  baspartn  13553  bastg  13564  tgcl  13567  epttop  13593  difopn  13611  ssntr  13625  isopn3  13628  isopn3i  13638  neiuni  13664  resttopon  13674  restopn2  13686  ssidcn  13713  lmtopcnp  13753  txuni2  13759  hmeoimaf1o  13817  tgioo  14049  bj-elssuniab  14546
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