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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 5702 | . . 3 ⊢ Rel E | |
2 | 1 | relbrcnv 5973 | . 2 ⊢ (𝐴◡ E 𝐵 ↔ 𝐵 E 𝐴) |
3 | epelg 5469 | . 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 2113 class class class wbr 5069 E cep 5467 ◡ccnv 5557 |
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-ral 3146 df-rex 3147 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-br 5070 df-opab 5132 df-eprel 5468 df-xp 5564 df-rel 5565 df-cnv 5566 |
This theorem is referenced by: brcnvepres 35532 eccnvepres 35541 eleccnvep 35542 cnvepres 35559 rnxrncnvepres 35652 dfcoels 35679 br1cossincnvepres 35694 br1cossxrncnvepres 35696 dfeldisj5 35958 |
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