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Theorem nrexrmo 2583
Description: Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
Assertion
Ref Expression
nrexrmo (¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)

Proof of Theorem nrexrmo
StepHypRef Expression
1 pm2.21 582 . 2 (¬ ∃𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
2 rmo5 2582 . 2 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
31, 2sylibr 132 1 (¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wrex 2360  ∃!wreu 2361  ∃*wrmo 2362
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 580
This theorem depends on definitions:  df-bi 115  df-mo 1952  df-rex 2365  df-reu 2366  df-rmo 2367
This theorem is referenced by: (None)
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