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Theorem intid 4207
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 4169 . . 3 {𝐴} ∈ V
3 eleq2 2234 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
41snid 3612 . . . 4 𝐴 ∈ {𝐴}
53, 4intmin3 3856 . . 3 ({𝐴} ∈ V → {𝑥𝐴𝑥} ⊆ {𝐴})
62, 5ax-mp 5 . 2 {𝑥𝐴𝑥} ⊆ {𝐴}
71elintab 3840 . . . 4 (𝐴 {𝑥𝐴𝑥} ↔ ∀𝑥(𝐴𝑥𝐴𝑥))
8 id 19 . . . 4 (𝐴𝑥𝐴𝑥)
97, 8mpgbir 1446 . . 3 𝐴 {𝑥𝐴𝑥}
10 snssi 3722 . . 3 (𝐴 {𝑥𝐴𝑥} → {𝐴} ⊆ {𝑥𝐴𝑥})
119, 10ax-mp 5 . 2 {𝐴} ⊆ {𝑥𝐴𝑥}
126, 11eqssi 3163 1 {𝑥𝐴𝑥} = {𝐴}
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141  {cab 2156  Vcvv 2730  wss 3121  {csn 3581   cint 3829
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-int 3830
This theorem is referenced by: (None)
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