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Theorem intid 3960
 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 intid.1 . . . 4
2 snexgOLD 3935 . . . 4
31, 2ax-mp 7 . . 3
4 eleq2 2101 . . . 4
51snid 3402 . . . 4
64, 5intmin3 3642 . . 3
73, 6ax-mp 7 . 2
81elintab 3626 . . . 4
9 id 19 . . . 4
108, 9mpgbir 1342 . . 3
11 snssi 3508 . . 3
1210, 11ax-mp 7 . 2
137, 12eqssi 2961 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1243   wcel 1393  cab 2026  cvv 2557   wss 2917  csn 3375  cint 3615 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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 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-pw 3361  df-sn 3381  df-int 3616 This theorem is referenced by: (None)
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