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Mirrors > Home > MPE Home > Th. List > dftr4 | Structured version Visualization version GIF version |
Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.) |
Ref | Expression |
---|---|
dftr4 | ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tr 5266 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
2 | sspwuni 5105 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | 1, 2 | bitr4i 278 | 1 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 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-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-ss 3980 df-pw 4607 df-uni 4913 df-tr 5266 |
This theorem is referenced by: tr0 5278 pwtr 5463 r1ordg 9816 r1sssuc 9821 r1val1 9824 ackbij2lem3 10278 tsktrss 10799 |
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