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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 5308 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | 1 | sseq1i 3943 | . 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
3 | df-tr 5137 | . 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴) | |
4 | dftr4 5141 | . 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) | |
5 | 2, 3, 4 | 3bitr4ri 307 | 1 ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 Tr wtr 5136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-uni 4801 df-tr 5137 |
This theorem is referenced by: r1tr 9189 itunitc1 9831 |
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