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Theorem 0pssin 39981
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 4399 . 2 (∅ ⊊ (𝐴𝐵) ↔ (𝐴𝐵) ≠ ∅)
2 ndisj 4331 . 2 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
31, 2bitri 276 1 (∅ ⊊ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1773  wcel 2107  wne 3021  cin 3939  wpss 3941  c0 4295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-v 3502  df-dif 3943  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296
This theorem is referenced by: (None)
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