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Mirrors > Home > MPE Home > Th. List > inn0f | Structured version Visualization version GIF version |
Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
inn0f.1 | ⊢ Ⅎ𝑥𝐴 |
inn0f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
inn0f | ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3978 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | 1 | exbii 1844 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
3 | inn0f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | inn0f.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | 3, 4 | nfin 4231 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
6 | 5 | n0f 4354 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵)) |
7 | df-rex 3068 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
8 | 2, 6, 7 | 3bitr4i 303 | 1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1775 ∈ wcel 2105 Ⅎwnfc 2887 ≠ wne 2937 ∃wrex 3067 ∩ cin 3961 ∅c0 4338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-rex 3068 df-v 3479 df-dif 3965 df-in 3969 df-nul 4339 |
This theorem is referenced by: inn0 4377 |
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