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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 33318 | . . 3 ⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E )) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ Trans ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) |
3 | eltrans.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3 | dftr6 32986 | . 2 ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) |
5 | 2, 4 | bitr4i 280 | 1 ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 Tr wtr 5172 E cep 5464 ran crn 5556 ∘ ccom 5559 Trans ctrans 33294 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-trans 33318 |
This theorem is referenced by: dfon3 33353 |
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