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Theorem n0rex 4082
 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 538 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
32eximi 1910 . 2 (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
4 n0 4078 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 df-rex 3067 . 2 (∃𝑥𝐴 𝑥𝐴 ↔ ∃𝑥(𝑥𝐴𝑥𝐴))
63, 4, 53imtr4i 281 1 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382  ∃wex 1852   ∈ wcel 2145   ≠ wne 2943  ∃wrex 3062  ∅c0 4063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rex 3067  df-v 3353  df-dif 3726  df-nul 4064 This theorem is referenced by:  ssn0rex  4083
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