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Theorem n0rex 3933
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 574 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
32eximi 1761 . 2 (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
4 n0 3929 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 df-rex 2917 . 2 (∃𝑥𝐴 𝑥𝐴 ↔ ∃𝑥(𝑥𝐴𝑥𝐴))
63, 4, 53imtr4i 281 1 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1703  wcel 1989  wne 2793  wrex 2912  c0 3913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-rex 2917  df-v 3200  df-dif 3575  df-nul 3914
This theorem is referenced by:  ssn0rex  3934
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