Theorem rmoeq 3699

Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by NM, 29-Oct-2020.)

Assertion

Ref	Expression
rmoeq	⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rmoeq

Step	Hyp	Ref	Expression
1		moeq 3668	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
2	1	moani 2581	. 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)
3		df-rmo 3369	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
4	2, 3	mpbir 234	1 ⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃*wmo 2565  ∃*wrmo 3368

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 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2567  df-cleq 2755  df-rmo 3369

This theorem is used by:  nbusgredgeu  30004  srepdflem2  36127