Theorem rextru 3077

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 1545	. . . 4 ⊢ ⊤
2	1	biantru 530	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
3	2	exbii 1850	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
4		df-rex 3071	. 2 ⊢ (∃𝑥 ∈ 𝐴 ⊤ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
5	3, 4	bitr4i 277	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)

Syntax hints:   ↔ wb 205   ∧ wa 396  ⊤wtru 1542  ∃wex 1781   ∈ wcel 2106  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-rex 3071

This theorem is referenced by:  iinss2d  43836  ralfal  43840  reutruALT  47444