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| Mirrors > Home > MPE Home > Th. List > dftr3 | Structured version Visualization version GIF version | ||
| Description: An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.) |
| Ref | Expression |
|---|---|
| dftr3 | ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dftr5 5263 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
| 2 | dfss3 3972 | . . 3 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
| 3 | 2 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) |
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 Tr wtr 5259 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 |
| This theorem is referenced by: trss 5270 trin 5271 triun 5274 triin 5276 tron 6407 ssorduni 7799 sucexeloniOLD 7830 suceloniOLD 7832 dfrecs3 8412 dfrecs3OLD 8413 ordtypelem2 9559 tcwf 9923 itunitc 10461 wunex2 10778 wfgru 10856 nadd2rabtr 43397 trwf 44976 |
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