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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 3967 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
| 2 | 1 | exbii 1848 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
| 3 | inn0f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | inn0f.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 5 | 3, 4 | nfin 4224 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
| 6 | 5 | n0f 4349 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵)) |
| 7 | df-rex 3071 | . 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 1779 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 ∃wrex 3070 ∩ cin 3950 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-rex 3071 df-v 3482 df-dif 3954 df-in 3958 df-nul 4334 |
| This theorem is referenced by: inn0 4372 |
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