 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  exists1 Structured version   Visualization version   GIF version

Theorem exists1 2710
 Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4988. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exists1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem exists1
StepHypRef Expression
1 df-eu 2622 . 2 (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
2 equid 2097 . . . . . 6 𝑥 = 𝑥
32tbt 358 . . . . 5 (𝑥 = 𝑦 ↔ (𝑥 = 𝑦𝑥 = 𝑥))
4 bicom 212 . . . . 5 ((𝑥 = 𝑦𝑥 = 𝑥) ↔ (𝑥 = 𝑥𝑥 = 𝑦))
53, 4bitri 264 . . . 4 (𝑥 = 𝑦 ↔ (𝑥 = 𝑥𝑥 = 𝑦))
65albii 1895 . . 3 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥𝑥 = 𝑦))
76exbii 1924 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
8 nfae 2468 . . 3 𝑦𝑥 𝑥 = 𝑦
9819.9 2228 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
101, 7, 93bitr2i 288 1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∀wal 1629  ∃wex 1852  ∃!weu 2618 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-eu 2622 This theorem is referenced by:  exists2  2711
 Copyright terms: Public domain W3C validator