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Mirrors > Home > ILE Home > Th. List > 0disj | 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 3255 | . . 3 ⊢ ∅ ⊆ {𝑥} | |
2 | 1 | rgenw 2376 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} |
3 | sndisj 3760 | . 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | |
4 | disjss2 3748 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥 ∈ 𝐴 {𝑥} → Disj 𝑥 ∈ 𝐴 ∅)) | |
5 | 2, 3, 4 | mp2 16 | 1 ⊢ Disj 𝑥 ∈ 𝐴 ∅ |
Colors of variables: wff set class |
Syntax hints: ∀wral 2306 ⊆ wss 2917 ∅c0 3224 {csn 3375 Disj wdisj 3745 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rmo 2314 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-disj 3746 |
This theorem is referenced by: (None) |
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