Theorem rmo0 4355

Description: Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)

Assertion

Ref	Expression
rmo0	⊢ ∃*𝑥 ∈ ∅ 𝜑

Proof of Theorem rmo0

Step	Hyp	Ref	Expression
1		rex0 4353	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2	1	pm2.21i 119	. 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3		rmo5 3395	. 2 ⊢ (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
4	2, 3	mpbir 230	1 ⊢ ∃*𝑥 ∈ ∅ 𝜑

Syntax hints:   → wi 4  ∃wrex 3069  ∃!wreu 3373  ∃*wrmo 3374  ∅c0 4318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-dif 3947  df-nul 4319

This theorem is referenced by:  rmosn  4716