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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 35838 | . . 3 ⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E )) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ Trans ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) |
3 | eltrans.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3 | dftr6 35730 | . 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 2105 Vcvv 3477 ∖ cdif 3959 Tr wtr 5264 E cep 5587 ran crn 5689 ∘ ccom 5692 Trans ctrans 35814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-trans 35838 |
This theorem is referenced by: dfon3 35873 |
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