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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 4446 | . 2 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ≠ ∅) | |
2 | ndisj 4369 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bitri 274 | 1 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∃wex 1773 ∈ wcel 2098 ≠ wne 2936 ∩ cin 3946 ⊊ wpss 3948 ∅c0 4324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2937 df-v 3473 df-dif 3950 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 |
This theorem is referenced by: (None) |
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