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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 4845 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | 1 | sseq1i 3592 | . 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
3 | df-tr 4681 | . 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴) | |
4 | dftr4 4685 | . 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) | |
5 | 2, 3, 4 | 3bitr4ri 292 | 1 ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 Tr wtr 4680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-pr 4128 df-uni 4373 df-tr 4681 |
This theorem is referenced by: r1tr 8522 itunitc1 9125 |
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