NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  exists1 GIF version

Theorem exists1 2293
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 in set.mm. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exists1 (∃!x x = xx x = y)
Distinct variable group:   x,y

Proof of Theorem exists1
StepHypRef Expression
1 df-eu 2208 . 2 (∃!x x = xyx(x = xx = y))
2 equid 1676 . . . . . 6 x = x
32tbt 333 . . . . 5 (x = y ↔ (x = yx = x))
4 bicom 191 . . . . 5 ((x = yx = x) ↔ (x = xx = y))
53, 4bitri 240 . . . 4 (x = y ↔ (x = xx = y))
65albii 1566 . . 3 (x x = yx(x = xx = y))
76exbii 1582 . 2 (yx x = yyx(x = xx = y))
8 nfae 1954 . . 3 yx x = y
9819.9 1783 . 2 (yx x = yx x = y)
101, 7, 93bitr2i 264 1 (∃!x x = xx x = y)
Colors of variables: wff setvar class
Syntax hints:  wb 176  wal 1540  wex 1541   = wceq 1642  ∃!weu 2204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208
This theorem is referenced by:  exists2  2294
  Copyright terms: Public domain W3C validator