Theorem rmo0 4342

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 4340	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2	1	pm2.21i 119	. 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3		rmo5 3384	. 2 ⊢ (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
4	2, 3	mpbir 231	1 ⊢ ∃*𝑥 ∈ ∅ 𝜑

Syntax hints:   → wi 4  ∃wrex 3061  ∃!wreu 3362  ∃*wrmo 3363  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-dif 3934  df-nul 4314

This theorem is referenced by:  rmosn  4700