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Theorem elssuni 3635
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 2991 . 2  |-  A  C_  A
2 ssuni 3629 . 2  |-  ( ( A  C_  A  /\  A  e.  B )  ->  A  C_  U. B )
31, 2mpan 408 1  |-  ( A  e.  B  ->  A  C_ 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1409    C_ wss 2944   U.cuni 3607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-uni 3608
This theorem is referenced by:  unissel  3636  ssunieq  3640  pwuni  3970  pwel  3981  uniopel  4020  iunpw  4238  dmrnssfld  4622  fvssunirng  5217  relfvssunirn  5218  sefvex  5223  riotaexg  5499  pwuninel2  5927  tfrlem9  5965  tfrexlem  5978  unirnioo  8942  bj-elssuniab  10289
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