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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eltrans | Structured version Visualization version GIF version | ||
| Description: Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.) |
| Ref | Expression |
|---|---|
| eltrans.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| eltrans | ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-trans 35858 | . . 3 ⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E )) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ Trans ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) |
| 3 | eltrans.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 4 | 3 | dftr6 35751 | . 2 ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) |
| 5 | 2, 4 | bitr4i 278 | 1 ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 Tr wtr 5259 E cep 5583 ran crn 5686 ∘ ccom 5689 Trans ctrans 35834 |
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-trans 35858 |
| This theorem is referenced by: dfon3 35893 |
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