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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 4397 | . 2 ⊢ ∅ ⊆ {∅} | |
2 | disjxsn 5141 | . 2 ⊢ Disj 𝑥 ∈ {∅}𝐵 | |
3 | disjss1 5119 | . 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 3947 ∅c0 4323 {csn 4629 Disj wdisj 5113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-rmo 3373 df-v 3473 df-dif 3950 df-in 3954 df-ss 3964 df-nul 4324 df-sn 4630 df-disj 5114 |
This theorem is referenced by: (None) |
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