Theorem 0iin 4979

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 4914	. 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
2		vex 3497	. . . 4 ⊢ 𝑦 ∈ V
3		ral0 4455	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴
4	2, 3	2th 266	. . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
5	4	abbi2i 2953	. 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
6	1, 5	eqtr4i 2847	1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V

Syntax hints:   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  Vcvv 3494  ∅c0 4290  ∩ ciin 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1536  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ral 3143  df-v 3496  df-dif 3938  df-nul 4291  df-iin 4914

This theorem is referenced by:  iinrab2  4984  iinvdif  4994  riin0  4996  iin0  5253  xpriindi  5701  cmpfi  22010  ptbasfi  22183  pol0N  37039