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Theorem intid 4710
 Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
Hypothesis
Ref Expression
intid.1
Assertion
Ref Expression
intid
Distinct variable group:   ,

Proof of Theorem intid
StepHypRef Expression
1 snex 4693 . . 3
2 eleq2 2530 . . . 4
3 intid.1 . . . . 5
43snid 4057 . . . 4
52, 4intmin3 4315 . . 3
61, 5ax-mp 5 . 2
73elintab 4297 . . . 4
8 id 22 . . . 4
97, 8mpgbir 1622 . . 3
10 snssi 4174 . . 3
119, 10ax-mp 5 . 2
126, 11eqssi 3519 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  {cab 2442   cvv 3109  C_wss 3475  {csn 4029  |^|cint 4286 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-pr 4032  df-int 4287
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