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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 4172 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | 1 | exbii 1847 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
3 | inn0f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | inn0f.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | 3, 4 | nfin 4196 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
6 | 5 | n0f 4310 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵)) |
7 | df-rex 3147 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
8 | 2, 6, 7 | 3bitr4i 305 | 1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1779 ∈ wcel 2113 Ⅎwnfc 2964 ≠ wne 3019 ∃wrex 3142 ∩ cin 3938 ∅c0 4294 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-in 3946 df-nul 4295 |
This theorem is referenced by: inn0 41343 |
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