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Mirrors > Home > MPE Home > Th. List > dftr4 | Structured version Visualization version 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 5064 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
2 | sspwuni 4921 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | 1, 2 | bitr4i 279 | 1 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ⊆ wss 3859 𝒫 cpw 4453 ∪ cuni 4745 Tr wtr 5063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-v 3439 df-in 3866 df-ss 3874 df-pw 4455 df-uni 4746 df-tr 5064 |
This theorem is referenced by: tr0 5074 pwtr 5237 r1ordg 9053 r1sssuc 9058 r1val1 9061 ackbij2lem3 9509 tsktrss 10029 |
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