Theorem rextru 3070

Description: Two ways of expressing that a class has at least one element. (Contributed by Zhi Wang, 23-Sep-2024.)

Assertion

Ref	Expression
rextru	⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)

Proof of Theorem rextru

Step	Hyp	Ref	Expression
1		tru 1551	. . . 4 ⊢ ⊤
2	1	biantru 534	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
3	2	exbii 1855	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
4		df-rex 3064	. 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  ∃wex 1786   ∈ wcel 2119  ∃wrex 3063

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

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-rex 3064

This theorem is referenced by:  iinss2d  45604  ralfal  45608  reutruALT  49295