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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 4352 | . . 3 ⊢ ∅ ⊆ {𝑥} | |
2 | 1 | rgenw 3152 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} |
3 | sndisj 5059 | . 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | |
4 | disjss2 5036 | . 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 3140 ⊆ wss 3938 ∅c0 4293 {csn 4569 Disj wdisj 5033 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rmo 3148 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-disj 5034 |
This theorem is referenced by: (None) |
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