Theorem degenmgm 19130

Description: A degenerate magma: although the operation is not defined for all pairs of elements of the base set ((∅(+_g‘𝑀)1_o) and (∅(+_g‘𝑀)∅) are not defined, and therefore are ∅ by definition, which is contained in the base set), and its domain is not (a subset of) the base set (2_o is in the domain of the operation, but not in the base set) , the structure 𝑀 is still a magma according to our definition. (Contributed by AV, 18-Aug-2026.)

Hypothesis

Ref	Expression
degenmgm.m	⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}

Assertion

Ref	Expression
degenmgm	⊢ 𝑀 ∈ Mgm

Proof of Theorem degenmgm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5261	. . 3 ⊢ ∅ ∈ V
2	1	prid1 4723	. 2 ⊢ ∅ ∈ {∅, 1o}
3	2, 2	pm3.2i 476	. . . . 5 ⊢ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o})
4		1oelpr 8480	. . . . . 6 ⊢ 1o ∈ {∅, 1o}
5	4, 4	pm3.2i 476	. . . . 5 ⊢ (1o ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})
6	3, 5	pm3.2i 476	. . . 4 ⊢ ((∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}) ∧ (1o ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o}))
7		1oex 8479	. . . . . 6 ⊢ 1o ∈ V
8		oveq1 7425	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = (∅{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦))
9		df-ov 7421	. . . . . . . . 9 ⊢ (∅{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩)
10	8, 9	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩))
11	10	eleq1d 2846	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) ∈ {∅, 1o} ↔ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o}))
12	11	ralbidv 3186	. . . . . 6 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) ∈ {∅, 1o} ↔ ∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o}))
13		oveq1 7425	. . . . . . . . 9 ⊢ (𝑥 = 1o → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = (1o{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦))
14		df-ov 7421	. . . . . . . . 9 ⊢ (1o{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩)
15	13, 14	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩))
16	15	eleq1d 2846	. . . . . . 7 ⊢ (𝑥 = 1o → ((𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) ∈ {∅, 1o} ↔ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o}))
17	16	ralbidv 3186	. . . . . 6 ⊢ (𝑥 = 1o → (∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) ∈ {∅, 1o} ↔ ∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o}))
18	1, 7, 12, 17	ralpr 4661	. . . . 5 ⊢ (∀𝑥 ∈ {∅, 1o}∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) ∈ {∅, 1o} ↔ (∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ∧ ∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o}))
19		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ⟨∅, 𝑦⟩ = ⟨∅, ∅⟩)
20	19	fveq2d 6887	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, ∅⟩))
21		1n0 8488	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
22	21	necomi 3010	. . . . . . . . . . . 12 ⊢ ∅ ≠ 1o
23	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 1o)
24	1, 1	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 1o))
25	23, 24	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩
26	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 2o)
27	1, 1	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 2o))
28	26, 27	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩
29	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ ∅)
30	1, 1	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, ∅⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ ∅))
31	29, 30	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, ∅⟩
32		opex 5432	. . . . . . . . . . 11 ⊢ ⟨∅, ∅⟩ ∈ V
33	7, 7, 7, 32	fvtp0 7204	. . . . . . . . . 10 ⊢ ((⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩ ∧ ⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩ ∧ ⟨∅, ∅⟩ ≠ ⟨1o, ∅⟩) → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, ∅⟩) = ∅)
34	25, 28, 31, 33	mp3an 1490	. . . . . . . . 9 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, ∅⟩) = ∅
35	20, 34	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) = ∅)
36	35	eleq1d 2846	. . . . . . 7 ⊢ (𝑦 = ∅ → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
37		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨∅, 𝑦⟩ = ⟨∅, 1o⟩)
38	37	fveq2d 6887	. . . . . . . . 9 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 1o⟩))
39	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ 1o ≠ 1o)
40	1, 7	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨∅, 1o⟩ ≠ ⟨1o, 1o⟩ ↔ (∅ ≠ 1o ∨ 1o ≠ 1o))
41	39, 40	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 1o⟩
42	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ 1o ≠ 2o)
43	1, 7	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ 1o ≠ 2o))
44	42, 43	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩
45	21	olci 880	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ 1o ≠ ∅)
46	1, 7	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨∅, 1o⟩ ≠ ⟨1o, ∅⟩ ↔ (∅ ≠ 1o ∨ 1o ≠ ∅))
47	45, 46	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, ∅⟩
48		opex 5432	. . . . . . . . . . 11 ⊢ ⟨∅, 1o⟩ ∈ V
49	7, 7, 7, 48	fvtp0 7204	. . . . . . . . . 10 ⊢ ((⟨∅, 1o⟩ ≠ ⟨1o, 1o⟩ ∧ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩ ∧ ⟨∅, 1o⟩ ≠ ⟨1o, ∅⟩) → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 1o⟩) = ∅)
50	41, 44, 47, 49	mp3an 1490	. . . . . . . . 9 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 1o⟩) = ∅
51	38, 50	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) = ∅)
52	51	eleq1d 2846	. . . . . . 7 ⊢ (𝑦 = 1o → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
53	1, 7, 36, 52	ralpr 4661	. . . . . 6 ⊢ (∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}))
54		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ⟨1o, 𝑦⟩ = ⟨1o, ∅⟩)
55	54	fveq2d 6887	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, ∅⟩))
56	21	olci 880	. . . . . . . . . . 11 ⊢ (1o ≠ 1o ∨ 1o ≠ ∅)
57	7, 7	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨1o, 1o⟩ ≠ ⟨1o, ∅⟩ ↔ (1o ≠ 1o ∨ 1o ≠ ∅))
58	56, 57	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 1o⟩ ≠ ⟨1o, ∅⟩
59		2on0 8484	. . . . . . . . . . . 12 ⊢ 2o ≠ ∅
60	59	olci 880	. . . . . . . . . . 11 ⊢ (1o ≠ 1o ∨ 2o ≠ ∅)
61		2oex 8481	. . . . . . . . . . . 12 ⊢ 2o ∈ V
62	7, 61	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩ ↔ (1o ≠ 1o ∨ 2o ≠ ∅))
63	60, 62	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩
64		opex 5432	. . . . . . . . . . 11 ⊢ ⟨1o, ∅⟩ ∈ V
65	64, 7	fvtp3 7200	. . . . . . . . . 10 ⊢ ((⟨1o, 1o⟩ ≠ ⟨1o, ∅⟩ ∧ ⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩) → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, ∅⟩) = 1o)
66	58, 63, 65	mp2an 705	. . . . . . . . 9 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, ∅⟩) = 1o
67	55, 66	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) = 1o)
68	67	eleq1d 2846	. . . . . . 7 ⊢ (𝑦 = ∅ → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o} ↔ 1o ∈ {∅, 1o}))
69		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨1o, 𝑦⟩ = ⟨1o, 1o⟩)
70	69	fveq2d 6887	. . . . . . . . 9 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 1o⟩))
71		1one2o 8648	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
72	71	olci 880	. . . . . . . . . . 11 ⊢ (1o ≠ 1o ∨ 1o ≠ 2o)
73	7, 7	opthne 5451	. . . . . . . . . . 11 ⊢ (⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (1o ≠ 1o ∨ 1o ≠ 2o))
74	72, 73	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩
75		opex 5432	. . . . . . . . . . 11 ⊢ ⟨1o, 1o⟩ ∈ V
76	75, 7	fvtp1 7198	. . . . . . . . . 10 ⊢ ((⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩ ∧ ⟨1o, 1o⟩ ≠ ⟨1o, ∅⟩) → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 1o⟩) = 1o)
77	74, 58, 76	mp2an 705	. . . . . . . . 9 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 1o⟩) = 1o
78	70, 77	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) = 1o)
79	78	eleq1d 2846	. . . . . . 7 ⊢ (𝑦 = 1o → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o} ↔ 1o ∈ {∅, 1o}))
80	1, 7, 68, 79	ralpr 4661	. . . . . 6 ⊢ (∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o} ↔ (1o ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o}))
81	53, 80	anbi12i 640	. . . . 5 ⊢ ((∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ∧ ∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o}) ↔ ((∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}) ∧ (1o ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})))
82	18, 81	bitri 278	. . . 4 ⊢ (∀𝑥 ∈ {∅, 1o}∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) ∈ {∅, 1o} ↔ ((∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}) ∧ (1o ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})))
83	6, 82	mpbir 234	. . 3 ⊢ ∀𝑥 ∈ {∅, 1o}∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) ∈ {∅, 1o}
84		degenmgm.m	. . . . . 6 ⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}
85		eqid 2761	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
86	84, 85	degenmgmbas 19128	. . . . 5 ⊢ (Base‘𝑀) = {∅, 1o}
87	86	eqcomi 2770	. . . 4 ⊢ {∅, 1o} = (Base‘𝑀)
88		tpex 7760	. . . . 5 ⊢ {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩} ∈ V
89	84	grpplusg 17454	. . . . 5 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩} ∈ V → {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩} = (+g‘𝑀))
90	88, 89	ax-mp 5	. . . 4 ⊢ {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩} = (+g‘𝑀)
91	87, 90	ismgmn0 18811	. . 3 ⊢ (∅ ∈ {∅, 1o} → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ {∅, 1o}∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) ∈ {∅, 1o}))
92	83, 91	mpbiri 261	. 2 ⊢ (∅ ∈ {∅, 1o} → 𝑀 ∈ Mgm)
93	2, 92	ax-mp 5	1 ⊢ 𝑀 ∈ Mgm

This proof depends on syntax axioms:   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  Vcvv 3451  ∅c0 4279  {cpr 4586  {ctp 4588  ⟨cop 4590  ‘cfv 6537  (class class class)co 7418  1_oc1o 8462  2_oc2o 8463  ndxcnx 17364  Basecbs 17380  +_gcplusg 17421  Mgmcmgm 18807

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-n0 12600  df-z 12687  df-uz 12959  df-fz 13633  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-plusg 17434  df-mgm 18809

This theorem is used by: (None)