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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0pssin | Structured version Visualization version GIF version |
Description: Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.) |
Ref | Expression |
---|---|
0pssin | ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0pss 4452 | . 2 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ≠ ∅) | |
2 | ndisj 4375 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bitri 275 | 1 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 ∩ cin 3961 ⊊ wpss 3963 ∅c0 4338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-dif 3965 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 |
This theorem is referenced by: (None) |
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