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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 5693 | . . 3 ⊢ Rel E | |
2 | 1 | relbrcnv 5964 | . 2 ⊢ (𝐴◡ E 𝐵 ↔ 𝐵 E 𝐴) |
3 | epelg 5460 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 E 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
4 | 2, 3 | syl5bb 285 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 class class class wbr 5058 E cep 5458 ◡ccnv 5548 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-eprel 5459 df-xp 5555 df-rel 5556 df-cnv 5557 |
This theorem is referenced by: brcnvepres 35522 eccnvepres 35531 eleccnvep 35532 cnvepres 35549 rnxrncnvepres 35642 dfcoels 35669 br1cossincnvepres 35684 br1cossxrncnvepres 35686 dfeldisj5 35948 |
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