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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brcnvep | Structured version Visualization version GIF version |
Description: The converse of the binary epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.) |
Ref | Expression |
---|---|
brcnvep | ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rele 5840 | . . 3 ⊢ Rel E | |
2 | 1 | relbrcnv 6128 | . 2 ⊢ (𝐴◡ E 𝐵 ↔ 𝐵 E 𝐴) |
3 | epelg 5590 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 E 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
4 | 2, 3 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 class class class wbr 5148 E cep 5588 ◡ccnv 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: brcnvepres 38249 eccnvepres 38262 eleccnvep 38263 cnvepres 38280 rnxrncnvepres 38382 coss2cnvepres 38400 dfcoels 38412 br1cossincnvepres 38432 br1cossxrncnvepres 38434 dfeldisj5 38703 |
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