Theorem intmin2 3954

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 2877	. . 3 ⊢ {x ∈ V ∣ A ⊆ x} = {x ∣ A ⊆ x}
2	1	inteqi 3931	. 2 ⊢ ∩{x ∈ V ∣ A ⊆ x} = ∩{x ∣ A ⊆ x}
3		intmin2.1	. . 3 ⊢ A ∈ V
4		intmin 3947	. . 3 ⊢ (A ∈ V → ∩{x ∈ V ∣ A ⊆ x} = A)
5	3, 4	ax-mp 5	. 2 ⊢ ∩{x ∈ V ∣ A ⊆ x} = A
6	2, 5	eqtr3i 2375	1 ⊢ ∩{x ∣ A ⊆ x} = A

This proof depends on syntax axioms:   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619  Vcvv 2860   ⊆ wss 3258  ∩cint 3927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is used by: (None)