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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ablssgrpel | Structured version Visualization version GIF version |
Description: Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 19739. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ablssgrpel | ⊢ (𝐴 ∈ Abel → 𝐴 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ablssgrp 36808 | . 2 ⊢ Abel ⊆ Grp | |
2 | 1 | sseli 3969 | 1 ⊢ (𝐴 ∈ Abel → 𝐴 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Grpcgrp 18889 Abelcabl 19735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3465 df-in 3948 df-ss 3958 df-abl 19737 |
This theorem is referenced by: (None) |
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