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Theorem intid 4853
 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 snex 4835 . . 3 {𝐴} ∈ V
2 eleq2 2677 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
3 intid.1 . . . . 5 𝐴 ∈ V
43snid 4155 . . . 4 𝐴 ∈ {𝐴}
52, 4intmin3 4440 . . 3 ({𝐴} ∈ V → {𝑥𝐴𝑥} ⊆ {𝐴})
61, 5ax-mp 5 . 2 {𝑥𝐴𝑥} ⊆ {𝐴}
73elintab 4422 . . . 4 (𝐴 {𝑥𝐴𝑥} ↔ ∀𝑥(𝐴𝑥𝐴𝑥))
8 id 22 . . . 4 (𝐴𝑥𝐴𝑥)
97, 8mpgbir 1717 . . 3 𝐴 {𝑥𝐴𝑥}
10 snssi 4280 . . 3 (𝐴 {𝑥𝐴𝑥} → {𝐴} ⊆ {𝑥𝐴𝑥})
119, 10ax-mp 5 . 2 {𝐴} ⊆ {𝑥𝐴𝑥}
126, 11eqssi 3584 1 {𝑥𝐴𝑥} = {𝐴}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ⊆ wss 3540  {csn 4125  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-int 4411 This theorem is referenced by: (None)
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