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Theorem exmodc 2050
Description: If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
Assertion
Ref Expression
exmodc (DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))

Proof of Theorem exmodc
StepHypRef Expression
1 df-dc 821 . 2 (DECID𝑥𝜑 ↔ (∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2 pm2.21 607 . . . 4 (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
3 df-mo 2004 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
42, 3sylibr 133 . . 3 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
54orim2i 751 . 2 ((∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑) → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
61, 5sylbi 120 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 698  DECID wdc 820  wex 1469  ∃!weu 2000  ∃*wmo 2001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821  df-mo 2004
This theorem is referenced by: (None)
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