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Theorem wl-moae 35072
 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 1970 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 35073 and exists1 2723. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2598, trut 1544) 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 35071 . 2 (∃*𝑥⊤ → ∀𝑥 𝑥 = 𝑦)
2 hbaev 2064 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
3219.8w 1983 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥 𝑥 = 𝑦)
4 ax-1 6 . . . . . 6 (𝑥 = 𝑦 → (⊤ → 𝑥 = 𝑦))
54alimi 1813 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑥(⊤ → 𝑥 = 𝑦))
65eximi 1836 . . . 4 (∃𝑦𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(⊤ → 𝑥 = 𝑦))
73, 6syl 17 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(⊤ → 𝑥 = 𝑦))
8 df-mo 2598 . . 3 (∃*𝑥⊤ ↔ ∃𝑦𝑥(⊤ → 𝑥 = 𝑦))
97, 8sylibr 237 . 2 (∀𝑥 𝑥 = 𝑦 → ∃*𝑥⊤)
101, 9impbii 212 1 (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ⊤wtru 1539  ∃wex 1781  ∃*wmo 2596 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 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-mo 2598 This theorem is referenced by:  wl-euae  35073
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