Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE 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 | pm2.21 579 | . 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
2 | rmo5 2569 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
3 | 1, 2 | sylibr 132 | 1 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∃wrex 2349 ∃!wreu 2350 ∃*wrmo 2351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in2 577 |
This theorem depends on definitions: df-bi 115 df-mo 1945 df-rex 2354 df-reu 2355 df-rmo 2356 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |