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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 4316 | . 2 ⊢ ∅ ⊆ 𝒫 ∅ | |
2 | dftr4 5171 | . 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅) | |
3 | 1, 2 | mpbir 234 | 1 ⊢ Tr ∅ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3871 ∅c0 4242 𝒫 cpw 4518 Tr wtr 5166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-v 3415 df-dif 3874 df-in 3878 df-ss 3888 df-nul 4243 df-pw 4520 df-uni 4825 df-tr 5167 |
This theorem is referenced by: ord0 6270 tctr 9361 tc0 9368 r1tr 9397 |
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