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Theorem intid 4018
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 3987 . . 3 {𝐴} ∈ V
3 eleq2 2148 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
41snid 3452 . . . 4 𝐴 ∈ {𝐴}
53, 4intmin3 3692 . . 3 ({𝐴} ∈ V → {𝑥𝐴𝑥} ⊆ {𝐴})
62, 5ax-mp 7 . 2 {𝑥𝐴𝑥} ⊆ {𝐴}
71elintab 3676 . . . 4 (𝐴 {𝑥𝐴𝑥} ↔ ∀𝑥(𝐴𝑥𝐴𝑥))
8 id 19 . . . 4 (𝐴𝑥𝐴𝑥)
97, 8mpgbir 1385 . . 3 𝐴 {𝑥𝐴𝑥}
10 snssi 3558 . . 3 (𝐴 {𝑥𝐴𝑥} → {𝐴} ⊆ {𝑥𝐴𝑥})
119, 10ax-mp 7 . 2 {𝐴} ⊆ {𝑥𝐴𝑥}
126, 11eqssi 3028 1 {𝑥𝐴𝑥} = {𝐴}
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  {cab 2071  Vcvv 2614  wss 2986  {csn 3425   cint 3665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-int 3666
This theorem is referenced by: (None)
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