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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 4343 | . . 3 ⊢ ∅ ⊆ {𝑥} | |
2 | 1 | rgenw 3065 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} |
3 | sndisj 5083 | . 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | |
4 | disjss2 5060 | . 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 3061 ⊆ wss 3898 ∅c0 4269 {csn 4573 Disj wdisj 5057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-mo 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rmo 3349 df-v 3443 df-dif 3901 df-in 3905 df-ss 3915 df-nul 4270 df-sn 4574 df-disj 5058 |
This theorem is referenced by: (None) |
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