Theorem rmotru 47866

Description: Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
rmotru	⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤)

Proof of Theorem rmotru

Step	Hyp	Ref	Expression
1		tru 1538	. . . 4 ⊢ ⊤
2	1	biantru 529	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
3	2	mobii 2538	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
4		df-rmo 3372	. 2 ⊢ (∃*𝑥 ∈ 𝐴 ⊤ ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
5	3, 4	bitr4i 278	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤)

Syntax hints:   ↔ wb 205   ∧ wa 395  ⊤wtru 1535   ∈ wcel 2099  ∃*wmo 2528  ∃*wrmo 3371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-mo 2530  df-rmo 3372

This theorem is referenced by:  reutruALT  47868  mosn  47874