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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 5451 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | 1 | sseq1i 4011 | . 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
3 | df-tr 5267 | . 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴) | |
4 | dftr4 5273 | . 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 205 ⊆ wss 3949 𝒫 cpw 4603 ∪ cuni 4909 Tr wtr 5266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-un 3954 df-in 3956 df-ss 3966 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4910 df-tr 5267 |
This theorem is referenced by: r1tr 9771 itunitc1 10415 |
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