Theorem intidg 5436

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 5267. (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 5409	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
2		snidg 4624	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3		eleq2 2851	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
4	1, 2, 3	elabd 3638	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ {𝑥 ∣ 𝐴 ∈ 𝑥})
5		intss1 4926	. . 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 1828	. . . 4 ⊢ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9		elintabg 4921	. . . 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 4750	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
12	6, 11	eqssd 3951	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2740  Vcvv 3453   ⊆ wss 3902  {csn 4587  ∩ cint 4910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-int 4911

This theorem is used by: (None)