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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 34373 | . . 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 34118 | . 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 205 ∈ wcel 2107 Vcvv 3444 ∖ cdif 3906 Tr wtr 5221 E cep 5534 ran crn 5632 ∘ ccom 5635 Trans ctrans 34349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-tr 5222 df-eprel 5535 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-trans 34373 |
This theorem is referenced by: dfon3 34408 |
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