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Mirrors > Home > ILE Home > Th. List > exmodc | GIF version |
Description: If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
Ref | Expression |
---|---|
exmodc | ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 835 | . 2 ⊢ (DECID ∃𝑥𝜑 ↔ (∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | |
2 | pm2.21 617 | . . . 4 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
3 | df-mo 2030 | . . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
4 | 2, 3 | sylibr 134 | . . 3 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
5 | 4 | orim2i 761 | . 2 ⊢ ((∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑) → (∃𝑥𝜑 ∨ ∃*𝑥𝜑)) |
6 | 1, 5 | sylbi 121 | 1 ⊢ (DECID ∃𝑥𝜑 → (∃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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