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Theorem moel 3401
Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2156. (Revised by Wolf Lammen, 23-Nov-2024.)
Assertion
Ref Expression
moel (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem moel
StepHypRef Expression
1 eleq1w 2823 . . 3 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21mo4 2565 . 2 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
3 r2al 3194 . 2 (∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
42, 3bitr4i 278 1 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537  wcel 2107  ∃*wmo 2537  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-clel 2815  df-ral 3061
This theorem is referenced by:  disjnf  32584  oppcmndclem  48920  isthinc3  49095  isthincd2lem1  49099  termcbasmo  49154  arweuthinc  49187  arweutermc  49188  mndtcbas2  49235
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