Theorem uniinn0 32639

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 2938	. . . 4 ⊢ (¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ (𝑥 ∩ 𝐵) = ∅)
2	1	ralbii 3085	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅)
3		ralnex 3065	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅)
4		unissb 4871	. . . 4 ⊢ (∪ 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵))
5		disj2 4386	. . . 4 ⊢ ((∪ 𝐴 ∩ 𝐵) = ∅ ↔ ∪ 𝐴 ⊆ (V ∖ 𝐵))
6		disj2 4386	. . . . 5 ⊢ ((𝑥 ∩ 𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵))
7	6	ralbii 3085	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵))
8	4, 5, 7	3bitr4ri 305	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅ ↔ (∪ 𝐴 ∩ 𝐵) = ∅)
9	2, 3, 8	3bitr3i 302	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅ ↔ (∪ 𝐴 ∩ 𝐵) = ∅)
10	9	necon1abii 2982	1 ⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  ∪ cuni 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-v 3433  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4262  df-uni 4839

This theorem is referenced by:  locfinreflem  34024