Theorem nrexrmo 2693

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		pm2.21 617	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
2		rmo5 2692	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
3	1, 2	sylibr 134	1 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∃wrex 2456  ∃!wreu 2457  ∃*wrmo 2458

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

This theorem depends on definitions:  df-bi 117  df-mo 2030  df-rex 2461  df-reu 2462  df-rmo 2463

This theorem is referenced by: (None)