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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 4683 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
2 | dfss3 3558 | . . 3 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | ralbii 2963 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) |
4 | 1, 3 | bitr4i 266 | 1 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 Tr wtr 4680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-in 3547 df-ss 3554 df-uni 4373 df-tr 4681 |
This theorem is referenced by: trss 4689 trssOLD 4690 trin 4691 triun 4694 trint 4696 tron 5663 ssorduni 6877 suceloni 6905 dfrecs3 7356 ordtypelem2 8307 tcwf 8629 itunitc 9126 wunex2 9439 wfgru 9517 |
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