Theorem euae 2659

Description: Two ways to express "exactly one thing exists". To paraphrase the statement and explain the label: there Exists a Unique thing if and only if for All 𝑥, 𝑥 Equals some given (and disjoint) 𝑦. Both sides are false in set theory, see Theorems neutru 34692 and dtru 5381. (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant ⊤. (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies. (Revised by Wolf Lammen, 2-Mar-2023.)

Assertion

Ref	Expression
euae	⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦

Proof of Theorem euae

Step	Hyp	Ref	Expression
1		extru 1978	. . 3 ⊢ ∃𝑥⊤
2	1	biantrur 531	. 2 ⊢ (∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦) ↔ (∃𝑥⊤ ∧ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)))
3		hbaev 2061	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
4	3	19.8w 1981	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥 𝑥 = 𝑦)
5		hbnaev 2064	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
6		alnex 1782	. . . . . 6 ⊢ (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∃𝑦∀𝑥 𝑥 = 𝑦)
7	5, 6	sylib 217	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑦∀𝑥 𝑥 = 𝑦)
8	7	con4i 114	. . . 4 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
9	4, 8	impbii 208	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥 𝑥 = 𝑦)
10		trut 1546	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ (⊤ → 𝑥 = 𝑦))
11	10	albii 1820	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(⊤ → 𝑥 = 𝑦))
12	11	exbii 1849	. . 3 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
13	9, 12	bitri 274	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
14		eu3v 2568	. 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)))
15	2, 13, 14	3bitr4ri 303	1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1538  ⊤wtru 1541  ∃wex 1780  ∃!weu 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-mo 2538  df-eu 2567

This theorem is referenced by:  exists1  2660