Theorem 0iin 5022

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 4954	. 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
2		vex 3455	. . . 4 ⊢ 𝑦 ∈ V
3		ral0 4454	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴
4	2, 3	2th 267	. . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
5	4	eqabi 2896	. 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
6	1, 5	eqtr4i 2787	1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  Vcvv 3451  ∅c0 4279  ∩ ciin 4952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-v 3453  df-dif 3902  df-nul 4280  df-iin 4954

This theorem is used by:  iinrab2  5028  iinvdif  5040  riin0  5042  iin0  5324  xpriindi  5813  cmpfi  23726  ptbasfi  23900  pol0N  40966