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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 3022 | . . . 4 ⊢ (¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ (𝑥 ∩ 𝐵) = ∅) | |
2 | 1 | ralbii 3167 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = ∅) |
3 | ralnex 3238 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑥 ∩ 𝐵) ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅) | |
4 | unissb 4872 | . . . 4 ⊢ (∪ 𝐴 ⊆ (V ∖ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ (V ∖ 𝐵)) | |
5 | disj2 4409 | . . . 4 ⊢ ((∪ 𝐴 ∩ 𝐵) = ∅ ↔ ∪ 𝐴 ⊆ (V ∖ 𝐵)) | |
6 | disj2 4409 | . . . . 5 ⊢ ((𝑥 ∩ 𝐵) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝐵)) | |
7 | 6 | ralbii 3167 | . . . 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 3066 | 1 ⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 ∪ cuni 4840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-uni 4841 |
This theorem is referenced by: locfinreflem 31106 |
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