Theorem euae 2693

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 36843 and dtru 5421. (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 2002	. . 3 ⊢ ∃𝑥⊤
2	1	biantrur 539	. 2 ⊢ (∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦) ↔ (∃𝑥⊤ ∧ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)))
3		hbaev 2088	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
4	3	19.8w 2005	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥 𝑥 = 𝑦)
5		hbnaev 2091	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
6		alnex 1808	. . . . . 6 ⊢ (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∃𝑦∀𝑥 𝑥 = 𝑦)
7	5, 6	sylib 221	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑦∀𝑥 𝑥 = 𝑦)
8	7	con4i 115	. . . 4 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
9	4, 8	impbii 212	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥 𝑥 = 𝑦)
10		trut 1573	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ (⊤ → 𝑥 = 𝑦))
11	10	albii 1846	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(⊤ → 𝑥 = 𝑦))
12	11	exbii 1875	. . 3 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
13	9, 12	bitri 278	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))
14		eu3v 2604	. 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)))
15	2, 13, 14	3bitr4ri 307	1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565  ⊤wtru 1568  ∃wex 1806  ∃!weu 2602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-mo 2573  df-eu 2603

This theorem is referenced by:  exists1  2694