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Mirrors > Home > MPE Home > Th. List > Mathboxes > uniinn0 | Structured version Visualization version GIF version |
Description: Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.) |
Ref | Expression |
---|---|
uniinn0 | ⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2947 | . . . 4 ⊢ (¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ (𝑥 ∩ 𝐵) = ∅) | |
2 | 1 | ralbii 3091 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅) |
3 | ralnex 3164 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅) | |
4 | unissb 4869 | . . . 4 ⊢ (∪ 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵)) | |
5 | disj2 4388 | . . . 4 ⊢ ((∪ 𝐴 ∩ 𝐵) = ∅ ↔ ∪ 𝐴 ⊆ (V ∖ 𝐵)) | |
6 | disj2 4388 | . . . . 5 ⊢ ((𝑥 ∩ 𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵)) | |
7 | 6 | ralbii 3091 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵)) |
8 | 4, 5, 7 | 3bitr4ri 307 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅ ↔ (∪ 𝐴 ∩ 𝐵) = ∅) |
9 | 2, 3, 8 | 3bitr3i 304 | . 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅ ↔ (∪ 𝐴 ∩ 𝐵) = ∅) |
10 | 9 | necon1abii 2992 | 1 ⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1543 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 Vcvv 3423 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ∪ cuni 4835 |
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 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ne 2944 df-ral 3069 df-rex 3070 df-v 3425 df-dif 3886 df-in 3890 df-ss 3900 df-nul 4254 df-uni 4836 |
This theorem is referenced by: locfinreflem 31535 |
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