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Theorem moel 29451
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 3255 . 2 (∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
2 df-ral 2946 . . 3 (∀𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦𝐴𝑥 = 𝑦))
32ralbii 3009 . 2 (∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦))
4 alcom 2077 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
5 eleq1 2718 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65mo4 2546 . . 3 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
7 df-ral 2946 . . . . 5 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
8 impexp 461 . . . . . 6 (((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
98albii 1787 . . . . 5 (∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
107, 9bitr4i 267 . . . 4 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
1110albii 1787 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
124, 6, 113bitr4i 292 . 2 (∃*𝑥 𝑥𝐴 ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
131, 3, 123bitr4ri 293 1 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wcel 2030  ∃*wmo 2499  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233
This theorem is referenced by:  disjnf  29510
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