Theorem intid 4318

Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)

Hypothesis

Ref	Expression
intid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
intid	⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem intid

Step	Hyp	Ref	Expression
1		intid.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	snex 4277	. . 3 ⊢ {𝐴} ∈ V
3		eleq2 2294	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
4	1	snid 3701	. . . 4 ⊢ 𝐴 ∈ {𝐴}
5	3, 4	intmin3 3956	. . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
6	2, 5	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}
7	1	elintab 3940	. . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥))
8		id 19	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9	7, 8	mpgbir 1501	. . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
10		snssi 3818	. . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
11	9, 10	ax-mp 5	. 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
12	6, 11	eqssi 3242	1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2201  {cab 2216  Vcvv 2801   ⊆ wss 3199  {csn 3670  ∩ cint 3929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-int 3930

This theorem is referenced by: (None)