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Theorem uniinn0 28749
Description: Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
Assertion
Ref Expression
uniinn0 (( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem uniinn0
StepHypRef Expression
1 nne 2786 . . . 4 (¬ (𝑥𝐵) ≠ ∅ ↔ (𝑥𝐵) = ∅)
21ralbii 2963 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ∀𝑥𝐴 (𝑥𝐵) = ∅)
3 ralnex 2975 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
4 unissb 4405 . . . 4 ( 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
5 disj2 3976 . . . 4 (( 𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
6 disj2 3976 . . . . 5 ((𝑥𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵))
76ralbii 2963 . . . 4 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
84, 5, 73bitr4ri 292 . . 3 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ( 𝐴𝐵) = ∅)
92, 3, 83bitr3i 289 . 2 (¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅ ↔ ( 𝐴𝐵) = ∅)
109necon1abii 2830 1 (( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195   = wceq 1475  wne 2780  wral 2896  wrex 2897  Vcvv 3173  cdif 3537  cin 3539  wss 3540  c0 3874   cuni 4372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-uni 4373
This theorem is referenced by:  locfinreflem  29235
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