Theorem intmin2 3955

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 2823	. . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥}
2	1	inteqi 3933	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥}
3		intmin2.1	. . 3 ⊢ 𝐴 ∈ V
4		intmin 3949	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴)
5	3, 4	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴
6	2, 5	eqtr3i 2253	1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴

Syntax hints:   = wceq 1397   ∈ wcel 2201  {cab 2216  {crab 2513  Vcvv 2801   ⊆ wss 3199  ∩ 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-ext 2212

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-ral 2514  df-rab 2518  df-v 2803  df-in 3205  df-ss 3212  df-int 3930

This theorem is referenced by: (None)