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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 5177 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
2 | dfss3 3958 | . . 3 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | ralbii 3167 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) |
4 | 1, 3 | bitr4i 280 | 1 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 Tr wtr 5174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 df-in 3945 df-ss 3954 df-uni 4841 df-tr 5175 |
This theorem is referenced by: trss 5183 trin 5184 triun 5187 triin 5189 tron 6216 ssorduni 7502 suceloni 7530 dfrecs3 8011 ordtypelem2 8985 tcwf 9314 itunitc 9845 wunex2 10162 wfgru 10240 |
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