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Theorem rmotru 45729
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 1546 . . . 4
21biantru 533 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ ⊤))
32mobii 2549 . 2 (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥(𝑥𝐴 ∧ ⊤))
4 df-rmo 3062 . 2 (∃*𝑥𝐴 ⊤ ↔ ∃*𝑥(𝑥𝐴 ∧ ⊤))
53, 4bitr4i 281 1 (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥𝐴 ⊤)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wtru 1543  wcel 2114  ∃*wmo 2539  ∃*wrmo 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-mo 2541  df-rmo 3062
This theorem is referenced by:  reutruALT  45731  mosn  45737
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