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Theorem ssn0rex 3969
 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 3700 . 2 (𝐴𝐵 → (∃𝑥𝐴 𝑥𝐴 → ∃𝑥𝐵 𝑥𝐴))
2 n0rex 3968 . 2 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
31, 2impel 484 1 ((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2030   ≠ wne 2823  ∃wrex 2942   ⊆ wss 3607  ∅c0 3948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949 This theorem is referenced by:  uhgrvd00  26486
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