Theorem intmin2 1984

Description: Any set is the smallest of all sets that include it.

Hypothesis

Ref	Expression
intmin2.1	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem intmin2

Step	Hyp	Ref	Expression
1		intmin2.1	. . 3 ⊢ A ∈ V
2		intmin 1982	. . 3 ⊢ (A ∈ V → A = ∩{x ∈ V∣A ⊆ x})
3	1, 2	ax-mp 6	. 2 ⊢ A = ∩{x ∈ V∣A ⊆ x}
4		df-rab 1208	. . . 4 ⊢ {x ∈ V∣A ⊆ x} = {x∣(x ∈ V ∧ A ⊆ x)}
5		visset 1350	. . . . . 6 ⊢ x ∈ V
6	5	biantrur 544	. . . . 5 ⊢ (A ⊆ x ↔ (x ∈ V ∧ A ⊆ x))
7	6	biabi 1181	. . . 4 ⊢ {x∣A ⊆ x} = {x∣(x ∈ V ∧ A ⊆ x)}
8	4, 7	eqtr4 1122	. . 3 ⊢ {x ∈ V∣A ⊆ x} = {x∣A ⊆ x}
9	8	inteqi 1969	. 2 ⊢ ∩{x ∈ V∣A ⊆ x} = ∩{x∣A ⊆ x}
10	3, 9	eqtr 1119	1 ⊢ A = ∩{x∣A ⊆ x}

Syntax hints:   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∩cint 1965

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rab 1208  df-v 1349  df-in 1491  df-ss 1492  df-int 1966

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