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Theorem moel 28506
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 3101 . 2 (∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
2 df-ral 2805 . . 3 (∀𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦𝐴𝑥 = 𝑦))
32ralbii 2867 . 2 (∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦))
4 alcom 1974 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
5 eleq1 2580 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65mo4 2409 . . 3 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
7 df-ral 2805 . . . . 5 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
8 impexp 460 . . . . . 6 (((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
98albii 1722 . . . . 5 (∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
107, 9bitr4i 265 . . . 4 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
1110albii 1722 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
124, 6, 113bitr4i 290 . 2 (∃*𝑥 𝑥𝐴 ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
131, 3, 123bitr4ri 291 1 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wcel 1938  ∃*wmo 2363  wral 2800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-v 3079
This theorem is referenced by:  disjnf  28565
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