Theorem intid 3512

Description: The intersection of all sets to which a set belongs is the singleton of that set.

Hypothesis

Ref	Expression
intid.1	|- A e. _V

Assertion

Ref	Expression
intid	|- |^|{x | A e. x} = {A}

Distinct variable group:   x,A

Proof of Theorem intid

Step	Hyp	Ref	Expression
1		snex 3492	. . 3 |- {A} e. _V
2		eleq2 1958	. . . 4 |- (x = {A} -> (A e. x <-> A e. {A}))
3		intid.1	. . . . 5 |- A e. _V
4	3	snid 3069	. . . 4 |- A e. {A}
5	2, 4	intmin3 3245	. . 3 |- ({A} e. _V -> |^|{x | A e. x} C_ {A})
6	1, 5	ax-mp 7	. 2 |- |^|{x | A e. x} C_ {A}
7	3	elintab 3227	. . . 4 |- (A e. |^|{x | A e. x} <-> A.x(A e. x -> A e. x))
8		id 73	. . . 4 |- (A e. x -> A e. x)
9	7, 8	mpgbir 1334	. . 3 |- A e. |^|{x | A e. x}
10		snssi 3129	. . 3 |- (A e. |^|{x | A e. x} -> {A} C_ |^|{x | A e. x})
11	9, 10	ax-mp 7	. 2 |- {A} C_ |^|{x | A e. x}
12	6, 11	eqssi 2632	1 |- |^|{x | A e. x} = {A}

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593  {csn 3044  |^|cint 3214

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-int 3215

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