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Theorem moel 29899
 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 3426 . 2 (∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
2 df-ral 3095 . . 3 (∀𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦𝐴𝑥 = 𝑦))
32ralbii 3162 . 2 (∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦))
4 alcom 2152 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
5 eleq1w 2842 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65mo4 2585 . . 3 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
7 df-ral 3095 . . . . 5 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
8 impexp 443 . . . . . 6 (((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
98albii 1863 . . . . 5 (∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
107, 9bitr4i 270 . . . 4 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
1110albii 1863 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
124, 6, 113bitr4i 295 . 2 (∃*𝑥 𝑥𝐴 ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
131, 3, 123bitr4ri 296 1 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599   ∈ wcel 2107  ∃*wmo 2549  ∀wral 3090 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-v 3400 This theorem is referenced by:  disjnf  29951
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