Theorem intid 4926

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		snex 4908	. . 3 ⊢ {𝐴} ∈ V
2		eleq2 2690	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
3		intid.1	. . . . 5 ⊢ 𝐴 ∈ V
4	3	snid 4208	. . . 4 ⊢ 𝐴 ∈ {𝐴}
5	2, 4	intmin3 4505	. . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
6	1, 5	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}
7	3	elintab 4487	. . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥))
8		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9	7, 8	mpgbir 1726	. . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
10		snssi 4339	. . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
11	9, 10	ax-mp 5	. 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
12	6, 11	eqssi 3619	1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200   ⊆ wss 3574  {csn 4177  ∩ cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-int 4476

This theorem is referenced by: (None)