Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 3869 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | 1 | exbii 1855 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
3 | inn0f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | inn0f.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | 3, 4 | nfin 4117 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
6 | 5 | n0f 4243 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵)) |
7 | df-rex 3057 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
8 | 2, 6, 7 | 3bitr4i 306 | 1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1787 ∈ wcel 2112 Ⅎwnfc 2877 ≠ wne 2932 ∃wrex 3052 ∩ cin 3852 ∅c0 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-in 3860 df-nul 4224 |
This theorem is referenced by: inn0 42237 |
Copyright terms: Public domain | W3C validator |