Theorem uniinn0 32643

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 3087	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅)
3		ralnex 3067	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅)
4		unissb 4874	. . . 4 ⊢ (∪ 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵))
5		disj2 4389	. . . 4 ⊢ ((∪ 𝐴 ∩ 𝐵) = ∅ ↔ ∪ 𝐴 ⊆ (V ∖ 𝐵))
6		disj2 4389	. . . . 5 ⊢ ((𝑥 ∩ 𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵))
7	6	ralbii 3087	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵))
8	4, 5, 7	3bitr4ri 306	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅ ↔ (∪ 𝐴 ∩ 𝐵) = ∅)
9	2, 3, 8	3bitr3i 303	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅ ↔ (∪ 𝐴 ∩ 𝐵) = ∅)
10	9	necon1abii 2984	1 ⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1548   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  Vcvv 3433   ∖ cdif 3882   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  ∪ cuni 4841

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-v 3435  df-dif 3888  df-in 3892  df-ss 3902  df-nul 4265  df-uni 4842

This theorem is referenced by:  locfinreflem  34036