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Theorem ssn0rex 40109
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 3629 . 2 (𝐴𝐵 → (∃𝑥𝐴 𝑥𝐴 → ∃𝑥𝐵 𝑥𝐴))
2 n0rex 40108 . 2 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
31, 2impel 483 1 ((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  wne 2779  wrex 2896  wss 3539  c0 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-rex 2901  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874
This theorem is referenced by:  uhgrvd00  40745
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