Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > pwtr | 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 3972 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | 1 | sseq1i 3023 | . 2 ⊢ (∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
3 | df-tr 3876 | . 2 ⊢ (Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴) | |
4 | dftr4 3880 | . 2 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) | |
5 | 2, 3, 4 | 3bitr4ri 211 | 1 ⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ⊆ wss 2973 𝒫 cpw 3382 ∪ cuni 3601 Tr wtr 3875 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-uni 3602 df-tr 3876 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |