Theorem rmo0 4361

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 4359	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2	1	pm2.21i 119	. 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3		rmo5 3399	. 2 ⊢ (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
4	2, 3	mpbir 231	1 ⊢ ∃*𝑥 ∈ ∅ 𝜑

Syntax hints:   → wi 4  ∃wrex 3069  ∃!wreu 3377  ∃*wrmo 3378  ∅c0 4332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-dif 3953  df-nul 4333

This theorem is referenced by:  rmosn  4718