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Theorem intid 4146
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 {𝑥𝐴𝑥} = {𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem intid
StepHypRef Expression
1 intid.1 . . . 4 𝐴 ∈ V
21snex 4109 . . 3 {𝐴} ∈ V
3 eleq2 2203 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
41snid 3556 . . . 4 𝐴 ∈ {𝐴}
53, 4intmin3 3798 . . 3 ({𝐴} ∈ V → {𝑥𝐴𝑥} ⊆ {𝐴})
62, 5ax-mp 5 . 2 {𝑥𝐴𝑥} ⊆ {𝐴}
71elintab 3782 . . . 4 (𝐴 {𝑥𝐴𝑥} ↔ ∀𝑥(𝐴𝑥𝐴𝑥))
8 id 19 . . . 4 (𝐴𝑥𝐴𝑥)
97, 8mpgbir 1429 . . 3 𝐴 {𝑥𝐴𝑥}
10 snssi 3664 . . 3 (𝐴 {𝑥𝐴𝑥} → {𝐴} ⊆ {𝑥𝐴𝑥})
119, 10ax-mp 5 . 2 {𝐴} ⊆ {𝑥𝐴𝑥}
126, 11eqssi 3113 1 {𝑥𝐴𝑥} = {𝐴}
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  {cab 2125  Vcvv 2686  wss 3071  {csn 3527   cint 3771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-int 3772
This theorem is referenced by: (None)
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