Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uniinn0 Structured version   Visualization version   GIF version

Theorem uniinn0 29946
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 2973 . . . 4 (¬ (𝑥𝐵) ≠ ∅ ↔ (𝑥𝐵) = ∅)
21ralbii 3162 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ∀𝑥𝐴 (𝑥𝐵) = ∅)
3 ralnex 3174 . . 3 (∀𝑥𝐴 ¬ (𝑥𝐵) ≠ ∅ ↔ ¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
4 unissb 4706 . . . 4 ( 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
5 disj2 4250 . . . 4 (( 𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
6 disj2 4250 . . . . 5 ((𝑥𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵))
76ralbii 3162 . . . 4 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ∀𝑥𝐴 𝑥 ⊆ (V ∖ 𝐵))
84, 5, 73bitr4ri 296 . . 3 (∀𝑥𝐴 (𝑥𝐵) = ∅ ↔ ( 𝐴𝐵) = ∅)
92, 3, 83bitr3i 293 . 2 (¬ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅ ↔ ( 𝐴𝐵) = ∅)
109necon1abii 3017 1 (( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198   = wceq 1601  wne 2969  wral 3090  wrex 3091  Vcvv 3398  cdif 3789  cin 3791  wss 3792  c0 4141   cuni 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-v 3400  df-dif 3795  df-in 3799  df-ss 3806  df-nul 4142  df-uni 4674
This theorem is referenced by:  locfinreflem  30513
  Copyright terms: Public domain W3C validator