Theorem 0iin 4989

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 4924	. 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
2		vex 3499	. . . 4 ⊢ 𝑦 ∈ V
3		ral0 4458	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴
4	2, 3	2th 266	. . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
5	4	abbi2i 2955	. 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
6	1, 5	eqtr4i 2849	1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V

Syntax hints:   = wceq 1537   ∈ wcel 2114  {cab 2801  ∀wral 3140  Vcvv 3496  ∅c0 4293  ∩ ciin 4922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-ral 3145  df-v 3498  df-dif 3941  df-nul 4294  df-iin 4924

This theorem is referenced by:  iinrab2  4994  iinvdif  5004  riin0  5006  iin0  5263  xpriindi  5709  cmpfi  22018  ptbasfi  22191  pol0N  37047