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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 3461 | . . 3 ⊢ ∅ ⊆ {𝑥} | |
2 | 1 | rgenw 2532 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} |
3 | sndisj 3999 | . 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | |
4 | disjss2 3983 | . 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 2455 ⊆ wss 3129 ∅c0 3422 {csn 3592 Disj wdisj 3980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rmo 2463 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 df-sn 3598 df-disj 3981 |
This theorem is referenced by: (None) |
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