Theorem intmin2 3662

Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)

Hypothesis

Ref	Expression
intmin2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
intmin2	⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem intmin2

Step	Hyp	Ref	Expression
1		rabab 2620	. . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥}
2	1	inteqi 3640	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥}
3		intmin2.1	. . 3 ⊢ 𝐴 ∈ V
4		intmin 3656	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴)
5	3, 4	ax-mp 7	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴
6	2, 5	eqtr3i 2103	1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴

Syntax hints:   = wceq 1284   ∈ wcel 1433  {cab 2067  {crab 2352  Vcvv 2601   ⊆ wss 2973  ∩ cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-int 3637

This theorem is referenced by: (None)