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Theorem exists1 1993
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. (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 1900 . 2 (∃!x x = xyx(x = xx = y))
2 equid 1586 . . . . . 6 x = x
32tbt 236 . . . . 5 (x = y ↔ (x = yx = x))
4 bicom 128 . . . . 5 ((x = yx = x) ↔ (x = xx = y))
53, 4bitri 173 . . . 4 (x = y ↔ (x = xx = y))
65albii 1356 . . 3 (x x = yx(x = xx = y))
76exbii 1493 . 2 (yx x = yyx(x = xx = y))
8 hbae 1603 . . 3 (x x = yyx x = y)
9819.9h 1531 . 2 (yx x = yx x = y)
101, 7, 93bitr2i 197 1 (∃!x x = xx x = y)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240  wex 1378  ∃!weu 1897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-eu 1900
This theorem is referenced by:  exists2  1994
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