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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 𝑥𝐴 ↔ ∃𝑥𝐴 ⊤)

Proof of Theorem rextru
StepHypRef Expression
1 tru 1545 . . . 4
21biantru 530 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ ⊤))
32exbii 1850 . 2 (∃𝑥 𝑥𝐴 ↔ ∃𝑥(𝑥𝐴 ∧ ⊤))
4 df-rex 3071 . 2 (∃𝑥𝐴 ⊤ ↔ ∃𝑥(𝑥𝐴 ∧ ⊤))
53, 4bitr4i 277 1 (∃𝑥 𝑥𝐴 ↔ ∃𝑥𝐴 ⊤)
Colors of variables: wff setvar class
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
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