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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 33322 | . . 3 ⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E )) | |
2 | 1 | eleq2i 2907 | . 2 ⊢ (𝐴 ∈ Trans ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) |
3 | eltrans.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3 | dftr6 32990 | . 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 2113 Vcvv 3497 ∖ cdif 3936 Tr wtr 5175 E cep 5467 ran crn 5559 ∘ ccom 5562 Trans ctrans 33298 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-eprel 5468 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-trans 33322 |
This theorem is referenced by: dfon3 33357 |
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