MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0iin Structured version   Visualization version   GIF version

Theorem 0iin 4768
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 4713 . 2 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
2 vex 3388 . . . 4 𝑦 ∈ V
3 ral0 4269 . . . 4 𝑥 ∈ ∅ 𝑦𝐴
42, 32th 256 . . 3 (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦𝐴)
54abbi2i 2915 . 2 V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
61, 5eqtr4i 2824 1 𝑥 ∈ ∅ 𝐴 = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  {cab 2785  wral 3089  Vcvv 3385  c0 4115   ciin 4711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-dif 3772  df-nul 4116  df-iin 4713
This theorem is referenced by:  iinrab2  4773  iinvdif  4782  riin0  4784  iin0  5031  xpriindi  5462  cmpfi  21540  ptbasfi  21713  pol0N  35930
  Copyright terms: Public domain W3C validator