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Theorem ssn0rex 4167
 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 3892 . 2 (𝐴𝐵 → (∃𝑥𝐴 𝑥𝐴 → ∃𝑥𝐵 𝑥𝐴))
2 n0rex 4166 . 2 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
31, 2impel 501 1 ((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2164   ≠ wne 2999  ∃wrex 3118   ⊆ wss 3798  ∅c0 4146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-rex 3123  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812  df-nul 4147 This theorem is referenced by:  uhgrvd00  26839
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