Theorem intidg 5424

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 5259. (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 5397	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
2		snidg 4617	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3		eleq2 2849	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
4	1, 2, 3	elabd 3634	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ {𝑥 ∣ 𝐴 ∈ 𝑥})
5		intss1 4922	. . 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 1827	. . . 4 ⊢ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9		elintabg 4917	. . . 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 4743	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
12	6, 11	eqssd 3925	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450   ⊆ wss 3895  {csn 4584  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3896  df-un 4151  df-sn 4585  df-pr 4587  df-int 4907

This theorem is used by: (None)