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Theorem moel 3395
Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2147. (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 2812 . . 3 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21mo4 2556 . 2 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
3 r2al 3191 . 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 205  wa 395  wal 1532  wcel 2099  ∃*wmo 2528  wral 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-mo 2530  df-clel 2806  df-ral 3059
This theorem is referenced by:  disjnf  32359  isthinc3  48029  isthincd2lem1  48033  mndtcbas2  48095
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