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Theorem uniinn0 30309
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 3015 . . . 4 (¬ (𝑥𝐵) ≠ ∅ ↔ (𝑥𝐵) = ∅)
21ralbii 3157 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ∀𝑥𝐴 (𝑥𝐵) = ∅)
3 ralnex 3224 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
4 unissb 4845 . . . 4 ( 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
5 disj2 4379 . . . 4 (( 𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
6 disj2 4379 . . . . 5 ((𝑥𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵))
76ralbii 3157 . . . 4 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
84, 5, 73bitr4ri 307 . . 3 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ( 𝐴𝐵) = ∅)
92, 3, 83bitr3i 304 . 2 (¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅ ↔ ( 𝐴𝐵) = ∅)
109necon1abii 3059 1 (( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1538  wne 3011  wral 3130  wrex 3131  Vcvv 3469  cdif 3905  cin 3907  wss 3908  c0 4265   cuni 4813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4266  df-uni 4814
This theorem is referenced by:  locfinreflem  31162
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