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Theorem uniinn0 30640
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 2947 . . . 4 (¬ (𝑥𝐵) ≠ ∅ ↔ (𝑥𝐵) = ∅)
21ralbii 3091 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ∀𝑥𝐴 (𝑥𝐵) = ∅)
3 ralnex 3164 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
4 unissb 4869 . . . 4 ( 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
5 disj2 4388 . . . 4 (( 𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
6 disj2 4388 . . . . 5 ((𝑥𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵))
76ralbii 3091 . . . 4 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
84, 5, 73bitr4ri 307 . . 3 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ( 𝐴𝐵) = ∅)
92, 3, 83bitr3i 304 . 2 (¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅ ↔ ( 𝐴𝐵) = ∅)
109necon1abii 2992 1 (( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1543  wne 2943  wral 3064  wrex 3065  Vcvv 3423  cdif 3880  cin 3882  wss 3883  c0 4253   cuni 4835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2160  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-ral 3069  df-rex 3070  df-v 3425  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-uni 4836
This theorem is referenced by:  locfinreflem  31535
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