Theorem intid 3951

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 3926	. . . 4 ⊢ (A ∈ V → {A} ∈ V)
3	1, 2	ax-mp 7	. . 3 ⊢ {A} ∈ V
4		eleq2 2098	. . . 4 ⊢ (x = {A} → (A ∈ x ↔ A ∈ {A}))
5	1	snid 3394	. . . 4 ⊢ A ∈ {A}
6	4, 5	intmin3 3633	. . 3 ⊢ ({A} ∈ V → ∩ {x ∣ A ∈ x} ⊆ {A})
7	3, 6	ax-mp 7	. 2 ⊢ ∩ {x ∣ A ∈ x} ⊆ {A}
8	1	elintab 3617	. . . 4 ⊢ (A ∈ ∩ {x ∣ A ∈ x} ↔ ∀x(A ∈ x → A ∈ x))
9		id 19	. . . 4 ⊢ (A ∈ x → A ∈ x)
10	8, 9	mpgbir 1339	. . 3 ⊢ A ∈ ∩ {x ∣ A ∈ x}
11		snssi 3499	. . 3 ⊢ (A ∈ ∩ {x ∣ A ∈ x} → {A} ⊆ ∩ {x ∣ A ∈ x})
12	10, 11	ax-mp 7	. 2 ⊢ {A} ⊆ ∩ {x ∣ A ∈ x}
13	7, 12	eqssi 2955	1 ⊢ ∩ {x ∣ A ∈ x} = {A}

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {cab 2023  Vcvv 2551   ⊆ wss 2911  {csn 3367  ∩ cint 3606

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-int 3607

This theorem is referenced by: (None)