Theorem rmo0 4320

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 4318	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2	1	pm2.21i 120	. 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3		rmo5 3390	. 2 ⊢ (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
4	2, 3	mpbir 234	1 ⊢ ∃*𝑥 ∈ ∅ 𝜑

This proof depends on syntax axioms:   → wi 4  ∃wrex 3092  ∃!wreu 3370  ∃*wrmo 3371  ∅c0 4289

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-dif 3911  df-nul 4290

This theorem is used by:  rmosn  4690