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Theorem reutru 45730
Description: Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
Assertion
Ref Expression
reutru (∃!𝑥 𝑥𝐴 ↔ ∃!𝑥𝐴 ⊤)

Proof of Theorem reutru
StepHypRef Expression
1 tru 1546 . . . 4
21biantru 533 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ ⊤))
32eubii 2587 . 2 (∃!𝑥 𝑥𝐴 ↔ ∃!𝑥(𝑥𝐴 ∧ ⊤))
4 df-reu 3061 . 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  ∃!weu 2570  ∃!wreu 3056
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-eu 2571  df-reu 3061
This theorem is referenced by: (None)
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