Theorem intidg 5458

Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) Put in closed form and avoid ax-nul 5307. (Revised by BJ, 17-Jan-2025.)

Assertion

Ref	Expression
intidg	⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem intidg

Step	Hyp	Ref	Expression
1		snexg 5431	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
2		snidg 4663	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3		eleq2 2820	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
4	1, 2, 3	elabd 3672	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ {𝑥 ∣ 𝐴 ∈ 𝑥})
5		intss1 4968	. . 3 ⊢ ({𝐴} ∈ {𝑥 ∣ 𝐴 ∈ 𝑥} → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
6	4, 5	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
7		id 22	. . . . 5 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
8	7	ax-gen 1795	. . . 4 ⊢ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9		elintabg 4962	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)))
10	8, 9	mpbiri 257	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
11	10	snssd 4813	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
12	6, 11	eqssd 4000	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})

Syntax hints:   → wi 4  ∀wal 1537   = wceq 1539   ∈ wcel 2104  {cab 2707  Vcvv 3472   ⊆ wss 3949  {csn 4629  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-v 3474  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632  df-int 4952

This theorem is referenced by: (None)