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Theorem 0iin 4509
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 4453 . 2 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
2 vex 3176 . . . 4 𝑦 ∈ V
3 ral0 4028 . . . 4 𝑥 ∈ ∅ 𝑦𝐴
42, 32th 253 . . 3 (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦𝐴)
54abbi2i 2725 . 2 V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
61, 5eqtr4i 2635 1 𝑥 ∈ ∅ 𝐴 = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  {cab 2596  wral 2896  Vcvv 3173  c0 3874   ciin 4451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875  df-iin 4453
This theorem is referenced by:  iinrab2  4514  iinvdif  4523  riin0  4525  iin0  4760  xpriindi  5168  cmpfi  20969  ptbasfi  21142  pol0N  34007
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