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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 5256 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
2 | sspwuni 5093 | . 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 205 ⊆ wss 3940 𝒫 cpw 4594 ∪ cuni 4899 Tr wtr 5255 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-v 3468 df-in 3947 df-ss 3957 df-pw 4596 df-uni 4900 df-tr 5256 |
This theorem is referenced by: tr0 5268 pwtr 5442 r1ordg 9769 r1sssuc 9774 r1val1 9777 ackbij2lem3 10232 tsktrss 10752 |
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