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Theorem elssuni 3582
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 BA B)

Proof of Theorem elssuni
StepHypRef Expression
1 ssid 2941 . 2 AA
2 ssuni 3576 . 2 ((AA A B) → A B)
31, 2mpan 402 1 (A BA B)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  wss 2894   cuni 3554
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555
This theorem is referenced by:  unissel  3583  ssunieq  3587  pwuni  3917  pwel  3928  uniopel  3967  iunpw  4161  dmrnssfld  4522  fvssunirng  5115  relfvssunirn  5116  sefvex  5121  pwuninel2  5819  tfrlem9  5857  tfrexlem  5870  bj-elssuniab  7037
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