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Theorem ssn0rex 40311
 Description: There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)
Assertion
Ref Expression
ssn0rex ((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssn0rex
StepHypRef Expression
1 ssrexv 3630 . 2 (𝐴𝐵 → (∃𝑥𝐴 𝑥𝐴 → ∃𝑥𝐵 𝑥𝐴))
2 n0rex 40310 . 2 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
31, 2impel 484 1 ((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  uhgrvd00  40750
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