Theorem intid 3934

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	⊢ ∩ {x ∣ A ∈ x} = {A}

Distinct variable group:   x,A

Proof of Theorem intid

Step	Hyp	Ref	Expression
1		intid.1	. . . 4 ⊢ A ∈ V
2		snexgOLD 3909	. . . 4 ⊢ (A ∈ V → {A} ∈ V)
3	1, 2	ax-mp 7	. . 3 ⊢ {A} ∈ V
4		eleq2 2083	. . . 4 ⊢ (x = {A} → (A ∈ x ↔ A ∈ {A}))
5	1	snid 3377	. . . 4 ⊢ A ∈ {A}
6	4, 5	intmin3 3616	. . 3 ⊢ ({A} ∈ V → ∩ {x ∣ A ∈ x} ⊆ {A})
7	3, 6	ax-mp 7	. 2 ⊢ ∩ {x ∣ A ∈ x} ⊆ {A}
8	1	elintab 3600	. . . 4 ⊢ (A ∈ ∩ {x ∣ A ∈ x} ↔ ∀x(A ∈ x → A ∈ x))
9		id 19	. . . 4 ⊢ (A ∈ x → A ∈ x)
10	8, 9	mpgbir 1322	. . 3 ⊢ A ∈ ∩ {x ∣ A ∈ x}
11		snssi 3482	. . 3 ⊢ (A ∈ ∩ {x ∣ A ∈ x} → {A} ⊆ ∩ {x ∣ A ∈ x})
12	10, 11	ax-mp 7	. 2 ⊢ {A} ⊆ ∩ {x ∣ A ∈ x}
13	7, 12	eqssi 2938	1 ⊢ ∩ {x ∣ A ∈ x} = {A}

Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1374  {cab 2008  Vcvv 2535   ⊆ wss 2894  {csn 3350  ∩ cint 3589

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-int 3590

This theorem is referenced by: (None)