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Mirrors > Home > MPE Home > Th. List > 0disj | Structured version Visualization version GIF version |
Description: Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
0disj | ⊢ Disj 𝑥 ∈ 𝐴 ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4349 | . . 3 ⊢ ∅ ⊆ {𝑥} | |
2 | 1 | rgenw 3150 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} |
3 | sndisj 5056 | . 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | |
4 | disjss2 5033 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥 ∈ 𝐴 {𝑥} → Disj 𝑥 ∈ 𝐴 ∅)) | |
5 | 2, 3, 4 | mp2 9 | 1 ⊢ Disj 𝑥 ∈ 𝐴 ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∀wral 3138 ⊆ wss 3935 ∅c0 4290 {csn 4566 Disj wdisj 5030 |
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-mo 2618 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rmo 3146 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4567 df-disj 5031 |
This theorem is referenced by: (None) |
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