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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 4075 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
2 | sspwuni 3944 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | 1, 2 | bitr4i 186 | 1 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ⊆ wss 3111 𝒫 cpw 3553 ∪ cuni 3783 Tr wtr 4074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-in 3117 df-ss 3124 df-pw 3555 df-uni 3784 df-tr 4075 |
This theorem is referenced by: tr0 4085 pwtr 4191 pw1on 7173 |
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