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Theorem 0pssin 41360
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 4378 . 2 (∅ ⊊ (𝐴𝐵) ↔ (𝐴𝐵) ≠ ∅)
2 ndisj 4301 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
31, 2bitri 274 1 (∅ ⊊ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wex 1782  wcel 2106  wne 2943  cin 3885  wpss 3887  c0 4256
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3431  df-dif 3889  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257
This theorem is referenced by: (None)
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