Theorem intmin2 4975

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 3512	. . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥}
2	1	inteqi 4950	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥}
3		intmin2.1	. . 3 ⊢ 𝐴 ∈ V
4		intmin 4968	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴)
5	3, 4	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴
6	2, 5	eqtr3i 2767	1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2108  {cab 2714  {crab 3436  Vcvv 3480   ⊆ wss 3951  ∩ cint 4946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-ss 3968  df-int 4947

This theorem is referenced by:  dfid7  43625