Theorem rmo0 4309

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 4307	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
2	1	pm2.21i 120	. 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3		rmo5 3383	. 2 ⊢ (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
4	2, 3	mpbir 234	1 ⊢ ∃*𝑥 ∈ ∅ 𝜑

This proof depends on syntax axioms:   → wi 4  ∃wrex 3086  ∃!wreu 3363  ∃*wrmo 3364  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-dif 4047  df-nul 4279

This theorem is used by:  rmosn  4676