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Theorem exists2 1914
Description: A condition implying that at least two things exist. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
exists2 ((xφ x ¬ φ) → ¬ ∃!x x = x)

Proof of Theorem exists2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 hbeu1 1839 . . . . . 6 (∃!x x = xx∃!x x = x)
2 hba1 1436 . . . . . 6 (xφxxφ)
3 exists1 1913 . . . . . . 7 (∃!x x = xx x = y)
4 ax-16 1644 . . . . . . 7 (x x = y → (φxφ))
53, 4sylbi 113 . . . . . 6 (∃!x x = x → (φxφ))
61, 2, 5exlimd 1482 . . . . 5 (∃!x x = x → (xφxφ))
76com12 26 . . . 4 (xφ → (∃!x x = xxφ))
8 alex 1502 . . . 4 (xφ ↔ ¬ x ¬ φ)
97, 8syl6ib 149 . . 3 (xφ → (∃!x x = x → ¬ x ¬ φ))
109con2d 536 . 2 (xφ → (x ¬ φ → ¬ ∃!x x = x))
1110imp 114 1 ((xφ x ¬ φ) → ¬ ∃!x x = x)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 96  wal 1335  wex 1374   = wceq 1383  ∃!weu 1829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ia1 98  ax-ia2 99  ax-ia3 100  ax-in1 527  ax-in2 528  ax-io 607  ax-5 1336  ax-7 1338  ax-gen 1339  ax-ie1 1375  ax-ie2 1376  ax-8 1387  ax-10 1388  ax-11 1389  ax-i12 1391  ax-4 1392  ax-17 1402  ax-i9 1417  ax-ial 1430  ax-16 1644
This theorem depends on definitions:  df-bi 109  df-tru 1313  df-fal 1314  df-eu 1833
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