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Theorem ssn0rex 4316
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 4012 . 2 (𝐴𝐵 → (∃𝑥𝐴 𝑥𝐴 → ∃𝑥𝐵 𝑥𝐴))
2 n0rex 4315 . 2 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
31, 2impel 507 1 ((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wne 2940  wrex 3070  wss 3911  c0 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rex 3071  df-v 3446  df-dif 3914  df-in 3918  df-ss 3928  df-nul 4284
This theorem is referenced by:  uhgrvd00  28524
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