Theorem wl-moae 38199

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 1996 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 38200 and exists1 2687. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (dfmo 2567, trut 1575) 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 38198	. 2 ⊢ (∃*𝑥⊤ → ∀𝑥 𝑥 = 𝑦)
2		hbaev 2090	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
3	2	19.8w 2007	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥 𝑥 = 𝑦)
4		ax-1 6	. . . . . 6 ⊢ (𝑥 = 𝑦 → (⊤ → 𝑥 = 𝑦))
5	4	alimi 1840	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(⊤ → 𝑥 = 𝑦))
6	5	eximi 1864	. . . 4 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
7	3, 6	syl 18	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
8		dfmo 2567	. . 3 ⊢ (∃*𝑥⊤ ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
9	7, 8	sylibr 237	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃*𝑥⊤)
10	1, 9	impbii 212	1 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  ⊤wtru 1570  ∃wex 1808  ∃*wmo 2564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-mo 2566

This theorem is used by:  wl-euae  38200