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Theorem 0iin 4610
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 4555 . 2 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
2 vex 3234 . . . 4 𝑦 ∈ V
3 ral0 4109 . . . 4 𝑥 ∈ ∅ 𝑦𝐴
42, 32th 254 . . 3 (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦𝐴)
54abbi2i 2767 . 2 V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
61, 5eqtr4i 2676 1 𝑥 ∈ ∅ 𝐴 = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  c0 3948   ciin 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-nul 3949  df-iin 4555
This theorem is referenced by:  iinrab2  4615  iinvdif  4624  riin0  4626  iin0  4869  xpriindi  5291  cmpfi  21259  ptbasfi  21432  pol0N  35513
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