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Theorem 0iin 4713
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 4658 . 2 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
2 vex 3354 . . . 4 𝑦 ∈ V
3 ral0 4218 . . . 4 𝑥 ∈ ∅ 𝑦𝐴
42, 32th 254 . . 3 (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦𝐴)
54abbi2i 2887 . 2 V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
61, 5eqtr4i 2796 1 𝑥 ∈ ∅ 𝐴 = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  {cab 2757  wral 3061  Vcvv 3351  c0 4063   ciin 4656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3726  df-nul 4064  df-iin 4658
This theorem is referenced by:  iinrab2  4718  iinvdif  4727  riin0  4729  iin0  4971  xpriindi  5396  cmpfi  21432  ptbasfi  21605  pol0N  35716
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