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Mirrors > Home > MPE Home > Th. List > disjx0 | Structured version Visualization version GIF version |
Description: An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjx0 | ⊢ Disj 𝑥 ∈ ∅ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4005 | . 2 ⊢ ∅ ⊆ {∅} | |
2 | disjxsn 4678 | . 2 ⊢ Disj 𝑥 ∈ {∅}𝐵 | |
3 | disjss1 4658 | . 2 ⊢ (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵 → Disj 𝑥 ∈ ∅ 𝐵)) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ Disj 𝑥 ∈ ∅ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3607 ∅c0 3948 {csn 4210 Disj wdisj 4652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rmo 2949 df-v 3233 df-dif 3610 df-in 3614 df-ss 3621 df-nul 3949 df-sn 4211 df-disj 4653 |
This theorem is referenced by: (None) |
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