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

Theorem elssuni 3852
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 3190 . 2  |-  A  C_  A
2 ssuni 3846 . 2  |-  ( ( A  C_  A  /\  A  e.  B )  ->  A  C_  U. B )
31, 2mpan 424 1  |-  ( A  e.  B  ->  A  C_ 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2160    C_ wss 3144   U.cuni 3824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825
This theorem is referenced by:  unissel  3853  ssunieq  3857  pwuni  4207  pwel  4233  uniopel  4271  iunpw  4495  dmrnssfld  4905  iotaexab  5211  fvssunirng  5546  relfvssunirn  5547  sefvex  5552  riotaexg  5852  pwuninel2  6302  tfrlem9  6339  tfrexlem  6354  sbthlem1  6981  sbthlem2  6982  unirnioo  9998  eltopss  13946  toponss  13963  isbasis3g  13983  baspartn  13987  bastg  13998  tgcl  14001  epttop  14027  difopn  14045  ssntr  14059  isopn3  14062  isopn3i  14072  neiuni  14098  resttopon  14108  restopn2  14120  ssidcn  14147  lmtopcnp  14187  txuni2  14193  hmeoimaf1o  14251  tgioo  14483  bj-elssuniab  14981
  Copyright terms: Public domain W3C validator