Theorem 0iin 5021

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 4953	. 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
2		vex 3454	. . . 4 ⊢ 𝑦 ∈ V
3		ral0 4453	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴
4	2, 3	2th 267	. . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
5	4	eqabi 2895	. 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
6	1, 5	eqtr4i 2786	1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  Vcvv 3450  ∅c0 4277  ∩ ciin 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-dif 4046  df-nul 4278  df-iin 4953

This theorem is used by:  iinrab2  5027  iinvdif  5039  riin0  5041  iin0  5324  xpriindi  5814  cmpfi  23735  ptbasfi  23909  pol0N  41103