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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 4406 | . 2 ⊢ ∅ ⊆ 𝒫 ∅ | |
2 | dftr4 5272 | . 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅) | |
3 | 1, 2 | mpbir 231 | 1 ⊢ Tr ∅ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 Tr wtr 5265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-dif 3966 df-ss 3980 df-nul 4340 df-pw 4607 df-uni 4913 df-tr 5266 |
This theorem is referenced by: ord0 6439 tctr 9778 tc0 9785 r1tr 9814 |
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