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Theorem 0iin 5068
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 5001 . 2 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
2 vex 3479 . . . 4 𝑦 ∈ V
3 ral0 4513 . . . 4 𝑥 ∈ ∅ 𝑦𝐴
42, 32th 264 . . 3 (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦𝐴)
54eqabi 2870 . 2 V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
61, 5eqtr4i 2764 1 𝑥 ∈ ∅ 𝐴 = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  {cab 2710  wral 3062  Vcvv 3475  c0 4323   ciin 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-dif 3952  df-nul 4324  df-iin 5001
This theorem is referenced by:  iinrab2  5074  iinvdif  5084  riin0  5086  iin0  5361  xpriindi  5837  cmpfi  22912  ptbasfi  23085  pol0N  38780
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