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Mirrors > Home > MPE Home > Th. List > pwtr | Structured version Visualization version GIF version |
Description: A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.) |
Ref | Expression |
---|---|
pwtr | ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unipw 5461 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | 1 | sseq1i 4024 | . 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
3 | df-tr 5266 | . 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴) | |
4 | dftr4 5272 | . 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) | |
5 | 2, 3, 4 | 3bitr4ri 304 | 1 ⊢ (Tr 𝐴 ↔ Tr 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 ax-sep 5302 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-un 3968 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-tr 5266 |
This theorem is referenced by: r1tr 9814 itunitc1 10458 |
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