Theorem degenmgm 19050

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 5264	. . 3 ⊢ ∅ ∈ V
2	1	prid1 4723	. 2 ⊢ ∅ ∈ {∅, 1o}
3	2, 2	pm3.2i 476	. . . . 5 ⊢ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o})
4		1oelpr 8466	. . . . . 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 8465	. . . . . 6 ⊢ 1o ∈ V
8		oveq1 7420	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = (∅{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦))
9		df-ov 7416	. . . . . . . . 9 ⊢ (∅{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩)
10	8, 9	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩))
11	10	eleq1d 2845	. . . . . . 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 3185	. . . . . 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 7420	. . . . . . . . 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 7416	. . . . . . . . 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 2811	. . . . . . . 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 2845	. . . . . . 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 3185	. . . . . 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 6882	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, ∅⟩))
21		1n0 8474	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
22	21	necomi 3009	. . . . . . . . . . . 12 ⊢ ∅ ≠ 1o
23	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 1o)
24	1, 1	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 1o))
25	23, 24	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩
26	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 2o)
27	1, 1	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 2o))
28	26, 27	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩
29	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ ∅)
30	1, 1	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, ∅⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ ∅))
31	29, 30	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, ∅⟩
32		opex 5439	. . . . . . . . . . 11 ⊢ ⟨∅, ∅⟩ ∈ V
33	7, 7, 7, 32	fvtp0 7199	. . . . . . . . . 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 2811	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) = ∅)
36	35	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = ∅ → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
37		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨∅, 𝑦⟩ = ⟨∅, 1o⟩)
38	37	fveq2d 6882	. . . . . . . . 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 5458	. . . . . . . . . . 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 5458	. . . . . . . . . . 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 5458	. . . . . . . . . . 11 ⊢ (⟨∅, 1o⟩ ≠ ⟨1o, ∅⟩ ↔ (∅ ≠ 1o ∨ 1o ≠ ∅))
47	45, 46	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, ∅⟩
48		opex 5439	. . . . . . . . . . 11 ⊢ ⟨∅, 1o⟩ ∈ V
49	7, 7, 7, 48	fvtp0 7199	. . . . . . . . . 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 2811	. . . . . . . 8 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨∅, 𝑦⟩) = ∅)
52	51	eleq1d 2845	. . . . . . 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 6882	. . . . . . . . 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 5458	. . . . . . . . . . 11 ⊢ (⟨1o, 1o⟩ ≠ ⟨1o, ∅⟩ ↔ (1o ≠ 1o ∨ 1o ≠ ∅))
58	56, 57	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 1o⟩ ≠ ⟨1o, ∅⟩
59		2on0 8470	. . . . . . . . . . . 12 ⊢ 2o ≠ ∅
60	59	olci 880	. . . . . . . . . . 11 ⊢ (1o ≠ 1o ∨ 2o ≠ ∅)
61		2oex 8467	. . . . . . . . . . . 12 ⊢ 2o ∈ V
62	7, 61	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩ ↔ (1o ≠ 1o ∨ 2o ≠ ∅))
63	60, 62	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩
64		opex 5439	. . . . . . . . . . 11 ⊢ ⟨1o, ∅⟩ ∈ V
65	64, 7	fvtp3 7195	. . . . . . . . . 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 2811	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) = 1o)
68	67	eleq1d 2845	. . . . . . 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 6882	. . . . . . . . 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 8634	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
72	71	olci 880	. . . . . . . . . . 11 ⊢ (1o ≠ 1o ∨ 1o ≠ 2o)
73	7, 7	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (1o ≠ 1o ∨ 1o ≠ 2o))
74	72, 73	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩
75		opex 5439	. . . . . . . . . . 11 ⊢ ⟨1o, 1o⟩ ∈ V
76	75, 7	fvtp1 7193	. . . . . . . . . 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 2811	. . . . . . . 8 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}‘⟨1o, 𝑦⟩) = 1o)
79	78	eleq1d 2845	. . . . . . 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 2760	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
86	84, 85	degenmgmbas 19048	. . . . 5 ⊢ (Base‘𝑀) = {∅, 1o}
87	86	eqcomi 2769	. . . 4 ⊢ {∅, 1o} = (Base‘𝑀)
88		tpex 7747	. . . . 5 ⊢ {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩} ∈ V
89	84	grpplusg 17375	. . . . 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 18732	. . 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 2955  ∀wral 3076  Vcvv 3450  ∅c0 4279  {cpr 4586  {ctp 4588  ⟨cop 4590  ‘cfv 6533  (class class class)co 7413  1_oc1o 8448  2_oc2o 8449  ndxcnx 17285  Basecbs 17301  +_gcplusg 17342  Mgmcmgm 18728

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 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-n0 12529  df-z 12616  df-uz 12888  df-fz 13562  df-struct 17239  df-slot 17274  df-ndx 17286  df-base 17302  df-plusg 17355  df-mgm 18730

This theorem is used by: (None)