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Theorem exmodc 2076
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 835 . 2 (DECID𝑥𝜑 ↔ (∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2 pm2.21 617 . . . 4 (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
3 df-mo 2030 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
42, 3sylibr 134 . . 3 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
54orim2i 761 . 2 ((∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑) → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
61, 5sylbi 121 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 708  DECID wdc 834  wex 1492  ∃!weu 2026  ∃*wmo 2027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835  df-mo 2030
This theorem is referenced by: (None)
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