Theorem intmin2 4934

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 3480	. . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥}
2	1	inteqi 4910	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥}
3		intmin2.1	. . 3 ⊢ 𝐴 ∈ V
4		intmin 4927	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴)
5	3, 4	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴
6	2, 5	eqtr3i 2785	1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  {cab 2738  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-int 4907

This theorem is used by:  dfid7  44705