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Theorem elssuni 3764
 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 3117 . 2 𝐴𝐴
2 ssuni 3758 . 2 ((𝐴𝐴𝐴𝐵) → 𝐴 𝐵)
31, 2mpan 420 1 (𝐴𝐵𝐴 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1480   ⊆ wss 3071  ∪ cuni 3736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737 This theorem is referenced by:  unissel  3765  ssunieq  3769  pwuni  4116  pwel  4140  uniopel  4178  iunpw  4401  dmrnssfld  4802  fvssunirng  5436  relfvssunirn  5437  sefvex  5442  riotaexg  5734  pwuninel2  6179  tfrlem9  6216  tfrexlem  6231  sbthlem1  6845  sbthlem2  6846  unirnioo  9763  eltopss  12186  toponss  12203  isbasis3g  12223  baspartn  12227  bastg  12240  tgcl  12243  epttop  12269  difopn  12287  ssntr  12301  isopn3  12304  isopn3i  12314  neiuni  12340  resttopon  12350  restopn2  12362  ssidcn  12389  lmtopcnp  12429  txuni2  12435  hmeoimaf1o  12493  tgioo  12725  bj-elssuniab  13012
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