ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  0iin GIF version

Theorem 0iin 3818
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 3763 . 2 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
2 vex 2644 . . . 4 𝑦 ∈ V
3 ral0 3411 . . . 4 𝑥 ∈ ∅ 𝑦𝐴
42, 32th 173 . . 3 (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦𝐴)
54abbi2i 2214 . 2 V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
61, 5eqtr4i 2123 1 𝑥 ∈ ∅ 𝐴 = V
Colors of variables: wff set class
Syntax hints:   = wceq 1299  wcel 1448  {cab 2086  wral 2375  Vcvv 2641  c0 3310   ciin 3761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-dif 3023  df-nul 3311  df-iin 3763
This theorem is referenced by:  riin0  3831  iin0r  4033
  Copyright terms: Public domain W3C validator