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Mirrors > Home > MPE Home > Th. List > Mathboxes > inn0f | Structured version Visualization version GIF version |
Description: A non-empty 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 | inn0f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | inn0f.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfin 3782 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
4 | 3 | n0f 3886 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵)) |
5 | elin 3758 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
6 | 5 | exbii 1764 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
7 | df-rex 2902 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
8 | 7 | bicomi 213 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
9 | 4, 6, 8 | 3bitri 285 | 1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 Ⅎwnfc 2738 ≠ wne 2780 ∃wrex 2897 ∩ cin 3539 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-nul 3875 |
This theorem is referenced by: inn0 38270 |
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