Theorem uniinn0 32551

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	⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem uniinn0

Step	Hyp	Ref	Expression
1		nne 2940	. . . 4 ⊢ (¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ (𝑥 ∩ 𝐵) = ∅)
2	1	ralbii 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅)
3		ralnex 3068	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅)
4		unissb 4947	. . . 4 ⊢ (∪ 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵))
5		disj2 4464	. . . 4 ⊢ ((∪ 𝐴 ∩ 𝐵) = ∅ ↔ ∪ 𝐴 ⊆ (V ∖ 𝐵))
6		disj2 4464	. . . . 5 ⊢ ((𝑥 ∩ 𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵))
7	6	ralbii 3089	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵))
8	4, 5, 7	3bitr4ri 304	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅ ↔ (∪ 𝐴 ∩ 𝐵) = ∅)
9	2, 3, 8	3bitr3i 301	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅ ↔ (∪ 𝐴 ∩ 𝐵) = ∅)
10	9	necon1abii 2985	1 ⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1535   ≠ wne 2936  ∀wral 3057  ∃wrex 3066  Vcvv 3477   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  ∪ cuni 4915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ne 2937  df-ral 3058  df-rex 3067  df-v 3479  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-uni 4916

This theorem is referenced by:  locfinreflem  33764