Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rmotru Structured version   Visualization version   GIF version

Theorem rmotru 48723
Description: Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.)
Assertion
Ref Expression
rmotru (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥𝐴 ⊤)

Proof of Theorem rmotru
StepHypRef Expression
1 tru 1544 . . . 4
21biantru 529 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ ⊤))
32mobii 2548 . 2 (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥(𝑥𝐴 ∧ ⊤))
4 df-rmo 3380 . 2 (∃*𝑥𝐴 ⊤ ↔ ∃*𝑥(𝑥𝐴 ∧ ⊤))
53, 4bitr4i 278 1 (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥𝐴 ⊤)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wtru 1541  wcel 2108  ∃*wmo 2538  ∃*wrmo 3379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-mo 2540  df-rmo 3380
This theorem is referenced by:  reutruALT  48725  mosn  48732
  Copyright terms: Public domain W3C validator