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Theorem moel 30262
Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.)
Assertion
Ref Expression
moel (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem moel
StepHypRef Expression
1 ralcom4 3201 . 2 (∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
2 df-ral 3114 . . 3 (∀𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦𝐴𝑥 = 𝑦))
32ralbii 3136 . 2 (∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦))
4 alcom 2161 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
5 eleq1w 2875 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65mo4 2628 . . 3 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
7 df-ral 3114 . . . . 5 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
8 impexp 454 . . . . . 6 (((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
98albii 1821 . . . . 5 (∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
107, 9bitr4i 281 . . . 4 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
1110albii 1821 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
124, 6, 113bitr4i 306 . 2 (∃*𝑥 𝑥𝐴 ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
131, 3, 123bitr4ri 307 1 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wcel 2112  ∃*wmo 2599  wral 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-11 2159
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2601  df-clel 2873  df-ral 3114
This theorem is referenced by:  disjnf  30336
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