New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  n0moeu GIF version

Theorem n0moeu 3562
 Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
Assertion
Ref Expression
n0moeu (A → (∃*x x A∃!x x A))
Distinct variable group:   x,A

Proof of Theorem n0moeu
StepHypRef Expression
1 n0 3559 . . . 4 (Ax x A)
21biimpi 186 . . 3 (Ax x A)
32biantrurd 494 . 2 (A → (∃*x x A ↔ (x x A ∃*x x A)))
4 eu5 2242 . 2 (∃!x x A ↔ (x x A ∃*x x A))
53, 4syl6bbr 254 1 (A → (∃*x x A∃!x x A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205   ≠ wne 2516  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator