Theorem nrexrmo 3366

Description: Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)

Assertion

Ref	Expression
nrexrmo	⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Proof of Theorem nrexrmo

Step	Hyp	Ref	Expression
1		pm2.21 123	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
2		rmo5 3365	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
3	1, 2	sylibr 233	1 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∃wrex 3065  ∃!wreu 3066  ∃*wrmo 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540  df-eu 2569  df-rex 3070  df-rmo 3071  df-reu 3072

This theorem is referenced by: (None)