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Mirrors > Home > ILE Home > Th. List > dftr4 | GIF version |
Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.) |
Ref | Expression |
---|---|
dftr4 | ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tr 3855 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
2 | sspwuni 3739 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | 1, 2 | bitr4i 176 | 1 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ⊆ wss 2917 𝒫 cpw 3359 ∪ cuni 3580 Tr wtr 3854 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-in 2924 df-ss 2931 df-pw 3361 df-uni 3581 df-tr 3855 |
This theorem is referenced by: tr0 3865 pwtr 3955 |
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