ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  intid GIF version

Theorem intid 4242
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 4203 . . 3 {𝐴} ∈ V
3 eleq2 2253 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
41snid 3638 . . . 4 𝐴 ∈ {𝐴}
53, 4intmin3 3886 . . 3 ({𝐴} ∈ V → {𝑥𝐴𝑥} ⊆ {𝐴})
62, 5ax-mp 5 . 2 {𝑥𝐴𝑥} ⊆ {𝐴}
71elintab 3870 . . . 4 (𝐴 {𝑥𝐴𝑥} ↔ ∀𝑥(𝐴𝑥𝐴𝑥))
8 id 19 . . . 4 (𝐴𝑥𝐴𝑥)
97, 8mpgbir 1464 . . 3 𝐴 {𝑥𝐴𝑥}
10 snssi 3751 . . 3 (𝐴 {𝑥𝐴𝑥} → {𝐴} ⊆ {𝑥𝐴𝑥})
119, 10ax-mp 5 . 2 {𝐴} ⊆ {𝑥𝐴𝑥}
126, 11eqssi 3186 1 {𝑥𝐴𝑥} = {𝐴}
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  {cab 2175  Vcvv 2752  wss 3144  {csn 3607   cint 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-int 3860
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator