Theorem 0iin 4025

Description: An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)

Assertion

Ref	Expression
0iin	⊢ ∩x ∈ ∅ A = V

Proof of Theorem 0iin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3973	. 2 ⊢ ∩x ∈ ∅ A = {y ∣ ∀x ∈ ∅ y ∈ A}
2		vex 2863	. . . 4 ⊢ y ∈ V
3		ral0 3655	. . . 4 ⊢ ∀x ∈ ∅ y ∈ A
4	2, 3	2th 230	. . 3 ⊢ (y ∈ V ↔ ∀x ∈ ∅ y ∈ A)
5	4	abbi2i 2465	. 2 ⊢ V = {y ∣ ∀x ∈ ∅ y ∈ A}
6	1, 5	eqtr4i 2376	1 ⊢ ∩x ∈ ∅ A = V

Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  Vcvv 2860  ∅c0 3551  ∩ciin 3971

This theorem was proved from 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 theorem 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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iin 3973

This theorem is referenced by:  iinrab2  4030  riin0  4040