Theorem intidg 5440

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 5270. (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 5413	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
2		snidg 4627	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3		eleq2 2852	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
4	1, 2, 3	elabd 3641	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ {𝑥 ∣ 𝐴 ∈ 𝑥})
5		intss1 4929	. . 3 ⊢ ({𝐴} ∈ {𝑥 ∣ 𝐴 ∈ 𝑥} → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
6	4, 5	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
7		id 23	. . . . 5 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
8	7	ax-gen 1825	. . . 4 ⊢ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9		elintabg 4924	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)))
10	8, 9	mpbiri 261	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
11	10	snssd 4753	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
12	6, 11	eqssd 3955	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})

Syntax hints:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3906  {csn 4590  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-un 3911  df-ss 3923  df-sn 4591  df-pr 4593  df-int 4914

This theorem is referenced by: (None)