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Mirrors > Home > MPE Home > Th. List > tr0 | Structured version Visualization version GIF version |
Description: The empty set is transitive. (Contributed by NM, 16-Sep-1993.) |
Ref | Expression |
---|---|
tr0 | ⊢ Tr ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4398 | . 2 ⊢ ∅ ⊆ 𝒫 ∅ | |
2 | dftr4 5273 | . 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅) | |
3 | 1, 2 | mpbir 230 | 1 ⊢ Tr ∅ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 Tr wtr 5266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-v 3463 df-dif 3947 df-ss 3961 df-nul 4323 df-pw 4606 df-uni 4910 df-tr 5267 |
This theorem is referenced by: ord0 6424 tctr 9765 tc0 9772 r1tr 9801 |
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