Theorem 0iin 4768

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 4713	. 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
2		vex 3388	. . . 4 ⊢ 𝑦 ∈ V
3		ral0 4269	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴
4	2, 3	2th 256	. . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
5	4	abbi2i 2915	. 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
6	1, 5	eqtr4i 2824	1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V

Syntax hints:   = wceq 1653   ∈ wcel 2157  {cab 2785  ∀wral 3089  Vcvv 3385  ∅c0 4115  ∩ ciin 4711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-dif 3772  df-nul 4116  df-iin 4713

This theorem is referenced by:  iinrab2  4773  iinvdif  4782  riin0  4784  iin0  5031  xpriindi  5462  cmpfi  21540  ptbasfi  21713  pol0N  35930