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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 5139 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
2 | dfss3 3903 | . . 3 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | ralbii 3133 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) |
4 | 1, 3 | bitr4i 281 | 1 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 Tr wtr 5136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-11 2158 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-tr 5137 |
This theorem is referenced by: trss 5145 trin 5146 triun 5149 triin 5151 tron 6182 ssorduni 7480 suceloni 7508 dfrecs3 7992 ordtypelem2 8967 tcwf 9296 itunitc 9832 wunex2 10149 wfgru 10227 |
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