Theorem intid 4339

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 4297	. . 3 ⊢ {𝐴} ∈ V
3		eleq2 2296	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
4	1	snid 3719	. . . 4 ⊢ 𝐴 ∈ {𝐴}
5	3, 4	intmin3 3975	. . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
6	2, 5	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}
7	1	elintab 3959	. . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥))
8		id 19	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9	7, 8	mpgbir 1502	. . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
10		snssi 3837	. . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
11	9, 10	ax-mp 5	. 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
12	6, 11	eqssi 3253	1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  {cab 2218  Vcvv 2812   ⊆ wss 3210  {csn 3688  ∩ cint 3948

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-int 3949

This theorem is referenced by: (None)