ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elssuni GIF version

Theorem elssuni 3608
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 2964 . 2 𝐴𝐴
2 ssuni 3602 . 2 ((𝐴𝐴𝐴𝐵) → 𝐴 𝐵)
31, 2mpan 400 1 (𝐴𝐵𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1393  wss 2917   cuni 3580
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581
This theorem is referenced by:  unissel  3609  ssunieq  3613  pwuni  3943  pwel  3954  uniopel  3993  iunpw  4211  dmrnssfld  4595  fvssunirng  5190  relfvssunirn  5191  sefvex  5196  riotaexg  5472  pwuninel2  5897  tfrlem9  5935  tfrexlem  5948  unirnioo  8840  bj-elssuniab  9904
  Copyright terms: Public domain W3C validator