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Mirrors > Home > NFE Home > Th. List > nrexrmo | GIF version |
Description: Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.) |
Ref | Expression |
---|---|
nrexrmo | ⊢ (¬ ∃x ∈ A φ → ∃*x ∈ A φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.21 100 | . 2 ⊢ (¬ ∃x ∈ A φ → (∃x ∈ A φ → ∃!x ∈ A φ)) | |
2 | rmo5 2828 | . 2 ⊢ (∃*x ∈ A φ ↔ (∃x ∈ A φ → ∃!x ∈ A φ)) | |
3 | 1, 2 | sylibr 203 | 1 ⊢ (¬ ∃x ∈ A φ → ∃*x ∈ A φ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∃wrex 2616 ∃!wreu 2617 ∃*wrmo 2618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 df-mo 2209 df-rex 2621 df-reu 2622 df-rmo 2623 |
This theorem is referenced by: (None) |
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