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Theorem intid 5318
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 5300 . . 3 {𝐴} ∈ V
2 eleq2 2881 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
3 intid.1 . . . . 5 𝐴 ∈ V
43snid 4564 . . . 4 𝐴 ∈ {𝐴}
52, 4intmin3 4869 . . 3 ({𝐴} ∈ V → {𝑥𝐴𝑥} ⊆ {𝐴})
61, 5ax-mp 5 . 2 {𝑥𝐴𝑥} ⊆ {𝐴}
73elintab 4852 . . . 4 (𝐴 {𝑥𝐴𝑥} ↔ ∀𝑥(𝐴𝑥𝐴𝑥))
8 id 22 . . . 4 (𝐴𝑥𝐴𝑥)
97, 8mpgbir 1801 . . 3 𝐴 {𝑥𝐴𝑥}
10 snssi 4704 . . 3 (𝐴 {𝑥𝐴𝑥} → {𝐴} ⊆ {𝑥𝐴𝑥})
119, 10ax-mp 5 . 2 {𝐴} ⊆ {𝑥𝐴𝑥}
126, 11eqssi 3934 1 {𝑥𝐴𝑥} = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  {cab 2779  Vcvv 3444  wss 3884  {csn 4528   cint 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-int 4842
This theorem is referenced by: (None)
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