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Mirrors > Home > MPE Home > Th. List > euae | Structured version Visualization version GIF version |
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 34333 and dtru 5263. (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.) |
Ref | Expression |
---|---|
euae | ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | extru 1984 | . . 3 ⊢ ∃𝑥⊤ | |
2 | 1 | biantrur 534 | . 2 ⊢ (∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦) ↔ (∃𝑥⊤ ∧ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))) |
3 | hbaev 2065 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | |
4 | 3 | 19.8w 1987 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥 𝑥 = 𝑦) |
5 | hbnaev 2068 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦) | |
6 | alnex 1789 | . . . . . 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 114 | . . . 4 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦) |
9 | 4, 8 | impbii 212 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥 𝑥 = 𝑦) |
10 | trut 1549 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ (⊤ → 𝑥 = 𝑦)) | |
11 | 10 | albii 1827 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(⊤ → 𝑥 = 𝑦)) |
12 | 11 | exbii 1855 | . . 3 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)) |
13 | 9, 12 | bitri 278 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)) |
14 | eu3v 2569 | . 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦))) | |
15 | 2, 13, 14 | 3bitr4ri 307 | 1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 ⊤wtru 1544 ∃wex 1787 ∃!weu 2567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-mo 2539 df-eu 2568 |
This theorem is referenced by: exists1 2661 |
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