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Theorem intid 4423
 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 4407 . . 3
2 eleq2 2499 . . . 4
3 intid.1 . . . . 5
43snid 3843 . . . 4
52, 4intmin3 4080 . . 3
61, 5ax-mp 8 . 2
73elintab 4063 . . . 4
8 id 21 . . . 4
97, 8mpgbir 1560 . . 3
10 snssi 3944 . . 3
119, 10ax-mp 8 . 2
126, 11eqssi 3366 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  cab 2424  cvv 2958   wss 3322  csn 3816  cint 4052 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-pr 3823  df-int 4053
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