Theorem intidg 5398

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 5244. (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 5373	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
2		snidg 4613	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3		eleq2 2820	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
4	1, 2, 3	elabd 3637	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ {𝑥 ∣ 𝐴 ∈ 𝑥})
5		intss1 4913	. . 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 1796	. . . 4 ⊢ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9		elintabg 4908	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)))
10	8, 9	mpbiri 258	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
11	10	snssd 4761	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
12	6, 11	eqssd 3952	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436   ⊆ wss 3902  {csn 4576  ∩ cint 4897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-un 3907  df-ss 3919  df-sn 4577  df-pr 4579  df-int 4898

This theorem is referenced by: (None)