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

Theorem exists1 2545
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 4775. (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 2458 . 2 (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
2 equid 1925 . . . . . 6 𝑥 = 𝑥
32tbt 357 . . . . 5 (𝑥 = 𝑦 ↔ (𝑥 = 𝑦𝑥 = 𝑥))
4 bicom 210 . . . . 5 ((𝑥 = 𝑦𝑥 = 𝑥) ↔ (𝑥 = 𝑥𝑥 = 𝑦))
53, 4bitri 262 . . . 4 (𝑥 = 𝑦 ↔ (𝑥 = 𝑥𝑥 = 𝑦))
65albii 1736 . . 3 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥𝑥 = 𝑦))
76exbii 1763 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
8 nfae 2300 . . 3 𝑦𝑥 𝑥 = 𝑦
9819.9 2058 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
101, 7, 93bitr2i 286 1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wal 1472  wex 1694  ∃!weu 2454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-eu 2458
This theorem is referenced by:  exists2  2546
  Copyright terms: Public domain W3C validator