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Theorem elssuni 3649
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 3027 . 2  |-  A  C_  A
2 ssuni 3643 . 2  |-  ( ( A  C_  A  /\  A  e.  B )  ->  A  C_  U. B )
31, 2mpan 415 1  |-  ( A  e.  B  ->  A  C_ 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434    C_ wss 2982   U.cuni 3621
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988  df-ss 2995  df-uni 3622
This theorem is referenced by:  unissel  3650  ssunieq  3654  pwuni  3983  pwel  4001  uniopel  4039  iunpw  4257  dmrnssfld  4643  fvssunirng  5241  relfvssunirn  5242  sefvex  5247  riotaexg  5523  pwuninel2  5951  tfrlem9  5988  tfrexlem  6003  unirnioo  9124  bj-elssuniab  10861
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