Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  euae Structured version   Visualization version   GIF version

Theorem euae 2722
 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 33883 and dtru 5237. (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
StepHypRef Expression
1 extru 1980 . . 3 𝑥
21biantrur 534 . 2 (∃𝑦𝑥(⊤ → 𝑥 = 𝑦) ↔ (∃𝑥⊤ ∧ ∃𝑦𝑥(⊤ → 𝑥 = 𝑦)))
3 hbaev 2064 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
4319.8w 1983 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥 𝑥 = 𝑦)
5 hbnaev 2067 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
6 alnex 1783 . . . . . 6 (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∃𝑦𝑥 𝑥 = 𝑦)
75, 6sylib 221 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑦𝑥 𝑥 = 𝑦)
87con4i 114 . . . 4 (∃𝑦𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
94, 8impbii 212 . . 3 (∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦𝑥 𝑥 = 𝑦)
10 trut 1544 . . . . 5 (𝑥 = 𝑦 ↔ (⊤ → 𝑥 = 𝑦))
1110albii 1821 . . . 4 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(⊤ → 𝑥 = 𝑦))
1211exbii 1849 . . 3 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∃𝑦𝑥(⊤ → 𝑥 = 𝑦))
139, 12bitri 278 . 2 (∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦𝑥(⊤ → 𝑥 = 𝑦))
14 eu3v 2630 . 2 (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃𝑦𝑥(⊤ → 𝑥 = 𝑦)))
152, 13, 143bitr4ri 307 1 (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ⊤wtru 1539  ∃wex 1781  ∃!weu 2628 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  df-eu 2629 This theorem is referenced by:  exists1  2723
 Copyright terms: Public domain W3C validator