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Theorem n0rex 4356
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 1834 . 2 (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
4 n0 4352 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 df-rex 3070 . 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 1778  wcel 2107  wne 2939  wrex 3069  c0 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-ne 2940  df-rex 3070  df-dif 3953  df-nul 4333
This theorem is referenced by:  ssn0rex  4357  aks5lem7  42202
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