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Theorem elssuni 3636
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 (𝐴𝐵𝐴 𝐵)

Proof of Theorem elssuni
StepHypRef Expression
1 ssid 2992 . 2 𝐴𝐴
2 ssuni 3630 . 2 ((𝐴𝐴𝐴𝐵) → 𝐴 𝐵)
31, 2mpan 408 1 (𝐴𝐵𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  wss 2945   cuni 3608
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 2952  df-ss 2959  df-uni 3609
This theorem is referenced by:  unissel  3637  ssunieq  3641  pwuni  3971  pwel  3982  uniopel  4021  iunpw  4239  dmrnssfld  4623  fvssunirng  5218  relfvssunirn  5219  sefvex  5224  riotaexg  5500  pwuninel2  5928  tfrlem9  5966  tfrexlem  5979  unirnioo  8943  bj-elssuniab  10317
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