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Theorem intid 3934
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 A V
Assertion
Ref Expression
intid {xA x} = {A}
Distinct variable group:   x,A

Proof of Theorem intid
StepHypRef Expression
1 intid.1 . . . 4 A V
2 snexgOLD 3909 . . . 4 (A V → {A} V)
31, 2ax-mp 7 . . 3 {A} V
4 eleq2 2083 . . . 4 (x = {A} → (A xA {A}))
51snid 3377 . . . 4 A {A}
64, 5intmin3 3616 . . 3 ({A} V → {xA x} ⊆ {A})
73, 6ax-mp 7 . 2 {xA x} ⊆ {A}
81elintab 3600 . . . 4 (A {xA x} ↔ x(A xA x))
9 id 19 . . . 4 (A xA x)
108, 9mpgbir 1322 . . 3 A {xA x}
11 snssi 3482 . . 3 (A {xA x} → {A} ⊆ {xA x})
1210, 11ax-mp 7 . 2 {A} ⊆ {xA x}
137, 12eqssi 2938 1 {xA x} = {A}
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  {cab 2008  Vcvv 2535  wss 2894  {csn 3350   cint 3589
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-int 3590
This theorem is referenced by: (None)
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