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Theorem 0iin 4980
 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.
StepHypRef Expression
1 df-iin 4915 . 2 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
2 vex 3498 . . . 4 𝑦 ∈ V
3 ral0 4456 . . . 4 𝑥 ∈ ∅ 𝑦𝐴
42, 32th 266 . . 3 (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦𝐴)
54abbi2i 2953 . 2 V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
61, 5eqtr4i 2847 1 𝑥 ∈ ∅ 𝐴 = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  Vcvv 3495  ∅c0 4291  ∩ ciin 4913 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 3497  df-dif 3939  df-nul 4292  df-iin 4915 This theorem is referenced by:  iinrab2  4985  iinvdif  4995  riin0  4997  iin0  5254  xpriindi  5702  cmpfi  22010  ptbasfi  22183  pol0N  37039
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