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Theorem n0rex 4363
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 548 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
32eximi 1832 . 2 (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
4 n0 4359 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 df-rex 3069 . 2 (∃𝑥𝐴 𝑥𝐴 ↔ ∃𝑥(𝑥𝐴𝑥𝐴))
63, 4, 53imtr4i 292 1 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wcel 2106  wne 2938  wrex 3068  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-ne 2939  df-rex 3069  df-dif 3966  df-nul 4340
This theorem is referenced by:  ssn0rex  4364  aks5lem7  42182
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