![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > inn0 | Structured version Visualization version GIF version |
Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
inn0 | ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2902 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2902 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | inn0f 44062 | 1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 ∩ cin 3947 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-in 3955 df-nul 4323 |
This theorem is referenced by: qinioo 44547 |
Copyright terms: Public domain | W3C validator |