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Theorem intid 4060
 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 4026 . . 3 {𝐴} ∈ V
3 eleq2 2152 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
41snid 3479 . . . 4 𝐴 ∈ {𝐴}
53, 4intmin3 3721 . . 3 ({𝐴} ∈ V → {𝑥𝐴𝑥} ⊆ {𝐴})
62, 5ax-mp 7 . 2 {𝑥𝐴𝑥} ⊆ {𝐴}
71elintab 3705 . . . 4 (𝐴 {𝑥𝐴𝑥} ↔ ∀𝑥(𝐴𝑥𝐴𝑥))
8 id 19 . . . 4 (𝐴𝑥𝐴𝑥)
97, 8mpgbir 1388 . . 3 𝐴 {𝑥𝐴𝑥}
10 snssi 3587 . . 3 (𝐴 {𝑥𝐴𝑥} → {𝐴} ⊆ {𝑥𝐴𝑥})
119, 10ax-mp 7 . 2 {𝐴} ⊆ {𝑥𝐴𝑥}
126, 11eqssi 3042 1 {𝑥𝐴𝑥} = {𝐴}
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1290   ∈ wcel 1439  {cab 2075  Vcvv 2620   ⊆ wss 3000  {csn 3450  ∩ cint 3694 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-int 3695 This theorem is referenced by: (None)
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