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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 0mgm | Structured version Visualization version GIF version | ||
| Description: A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.) | 
| Ref | Expression | 
|---|---|
| 0mgm.b | ⊢ (Base‘𝑀) = ∅ | 
| Ref | Expression | 
|---|---|
| 0mgm | ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ral0 4513 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅ | |
| 2 | 0mgm.b | . . . 4 ⊢ (Base‘𝑀) = ∅ | |
| 3 | 2 | eqcomi 2746 | . . 3 ⊢ ∅ = (Base‘𝑀) | 
| 4 | eqid 2737 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 5 | 3, 4 | ismgm 18654 | . 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅)) | 
| 6 | 1, 5 | mpbiri 258 | 1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-mgm 18653 | 
| This theorem is referenced by: (None) | 
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