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Theorem wl-moae 33839
 Description: Two ways to express "at most one thing exists" or, in this context equivalently, "exactly one thing exists" . The equivalence results from the presence of ax-6 2075 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 33840 and exists1 2743. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2605, trut 1663) also means "exactly one thing exists" . (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant ⊤. (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies, and use ∃*. (Revised by Wolf Lammen, 7-Mar-2023.)
Assertion
Ref Expression
wl-moae (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem wl-moae
StepHypRef Expression
1 wl-motae 33838 . 2 (∃*𝑥⊤ → ∀𝑥 𝑥 = 𝑦)
2 hbaev 2159 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
3219.8w 2082 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥 𝑥 = 𝑦)
4 ax-1 6 . . . . . 6 (𝑥 = 𝑦 → (⊤ → 𝑥 = 𝑦))
54alimi 1910 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑥(⊤ → 𝑥 = 𝑦))
65eximi 1933 . . . 4 (∃𝑦𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(⊤ → 𝑥 = 𝑦))
73, 6syl 17 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(⊤ → 𝑥 = 𝑦))
8 df-mo 2605 . . 3 (∃*𝑥⊤ ↔ ∃𝑦𝑥(⊤ → 𝑥 = 𝑦))
97, 8sylibr 226 . 2 (∀𝑥 𝑥 = 𝑦 → ∃*𝑥⊤)
101, 9impbii 201 1 (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654  ⊤wtru 1657  ∃wex 1878  ∃*wmo 2603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112 This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1660  df-ex 1879  df-mo 2605 This theorem is referenced by:  wl-euae  33840
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