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Theorem uniinn0 29492
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 2827 . . . 4 (¬ (𝑥𝐵) ≠ ∅ ↔ (𝑥𝐵) = ∅)
21ralbii 3009 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ∀𝑥𝐴 (𝑥𝐵) = ∅)
3 ralnex 3021 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
4 unissb 4501 . . . 4 ( 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
5 disj2 4057 . . . 4 (( 𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
6 disj2 4057 . . . . 5 ((𝑥𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵))
76ralbii 3009 . . . 4 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
84, 5, 73bitr4ri 293 . . 3 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ( 𝐴𝐵) = ∅)
92, 3, 83bitr3i 290 . 2 (¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅ ↔ ( 𝐴𝐵) = ∅)
109necon1abii 2871 1 (( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1523  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-uni 4469
This theorem is referenced by:  locfinreflem  30035
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