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

Theorem rextru 3077
Description: Two ways of expressing that a class has at least one element. (Contributed by Zhi Wang, 23-Sep-2024.)
Assertion
Ref Expression
rextru (∃𝑥 𝑥𝐴 ↔ ∃𝑥𝐴 ⊤)

Proof of Theorem rextru
StepHypRef Expression
1 tru 1546 . . . 4
21biantru 531 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ ⊤))
32exbii 1851 . 2 (∃𝑥 𝑥𝐴 ↔ ∃𝑥(𝑥𝐴 ∧ ⊤))
4 df-rex 3071 . 2 (∃𝑥𝐴 ⊤ ↔ ∃𝑥(𝑥𝐴 ∧ ⊤))
53, 4bitr4i 278 1 (∃𝑥 𝑥𝐴 ↔ ∃𝑥𝐴 ⊤)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wtru 1543  wex 1782  wcel 2107  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-rex 3071
This theorem is referenced by:  iinss2d  43460  ralfal  43464  reutruALT  46977
  Copyright terms: Public domain W3C validator