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Theorem 0pssin 38846
Description: Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
Assertion
Ref Expression
0pssin (∅ ⊊ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem 0pssin
StepHypRef Expression
1 0pss 4209 . 2 (∅ ⊊ (𝐴𝐵) ↔ (𝐴𝐵) ≠ ∅)
2 ndisj 4146 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
31, 2bitri 267 1 (∅ ⊊ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wex 1875  wcel 2157  wne 2971  cin 3768  wpss 3770  c0 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116
This theorem is referenced by: (None)
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