Theorem rmotru 49293

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 1551	. . . 4 ⊢ ⊤
2	1	biantru 534	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
3	2	mobii 2552	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
4		df-rmo 3344	. 2 ⊢ (∃*𝑥 ∈ 𝐴 ⊤ ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
5	3, 4	bitr4i 279	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤)

Syntax hints:   ↔ wb 207   ∧ wa 396  ⊤wtru 1548   ∈ wcel 2119  ∃*wmo 2541  ∃*wrmo 3343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-mo 2543  df-rmo 3344

This theorem is referenced by:  reutruALT  49295  mosn  49303