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Theorem n0rex 4271
 Description: There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)
Assertion
Ref Expression
n0rex (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem n0rex
StepHypRef Expression
1 id 22 . . . 4 (𝑥𝐴𝑥𝐴)
21ancli 552 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
32eximi 1836 . 2 (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
4 n0 4263 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 df-rex 3115 . 2 (∃𝑥𝐴 𝑥𝐴 ↔ ∃𝑥(𝑥𝐴𝑥𝐴))
63, 4, 53imtr4i 295 1 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2112   ≠ wne 2990  ∃wrex 3110  ∅c0 4246 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-rex 3115  df-dif 3887  df-nul 4247 This theorem is referenced by:  ssn0rex  4272
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