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Mirrors > Home > ILE Home > Th. List > disjx0 | 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 3249 | . 2 ⊢ ∅ ⊆ {∅} | |
2 | disjxsn 3753 | . 2 ⊢ Disj x ∈ {∅}B | |
3 | disjss1 3742 | . 2 ⊢ (∅ ⊆ {∅} → (Disj x ∈ {∅}B → Disj x ∈ ∅ B)) | |
4 | 1, 2, 3 | mp2 16 | 1 ⊢ Disj x ∈ ∅ B |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 2911 ∅c0 3218 {csn 3367 Disj wdisj 3736 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rmo 2308 df-v 2553 df-dif 2914 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-disj 3737 |
This theorem is referenced by: (None) |
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