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

Step	Hyp	Ref	Expression
1		wl-motae 33838	. 2 ⊢ (∃*𝑥⊤ → ∀𝑥 𝑥 = 𝑦)
2		hbaev 2159	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
3	2	19.8w 2082	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥 𝑥 = 𝑦)
4		ax-1 6	. . . . . 6 ⊢ (𝑥 = 𝑦 → (⊤ → 𝑥 = 𝑦))
5	4	alimi 1910	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(⊤ → 𝑥 = 𝑦))
6	5	eximi 1933	. . . 4 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
7	3, 6	syl 17	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
8		df-mo 2605	. . 3 ⊢ (∃*𝑥⊤ ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
9	7, 8	sylibr 226	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃*𝑥⊤)
10	1, 9	impbii 201	1 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

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