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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 4395 | . 2 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ≠ ∅) | |
2 | ndisj 4326 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bitri 277 | 1 ⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∩ cin 3934 ⊊ wpss 3936 ∅c0 4290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 |
This theorem is referenced by: (None) |
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