Theorem pwmndid 18101

Description: The identity of the monoid of the power set of a class 𝐴 under union is the empty set. (Contributed by AV, 27-Feb-2024.)

Hypotheses

Ref	Expression
pwmnd.b	⊢ (Base‘𝑀) = 𝒫 𝐴
pwmnd.p	⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))

Assertion

Ref	Expression
pwmndid	⊢ (0g‘𝑀) = ∅

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑀(𝑥,𝑦)

Proof of Theorem pwmndid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 5256	. 2 ⊢ ∅ ∈ 𝒫 𝐴
2		pwmnd.b	. . . . 5 ⊢ (Base‘𝑀) = 𝒫 𝐴
3	2	eqcomi 2830	. . . 4 ⊢ 𝒫 𝐴 = (Base‘𝑀)
4		eqid 2821	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
5		eqid 2821	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
6		id 22	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ ∈ 𝒫 𝐴)
7		pwmnd.p	. . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))
8	2, 7	pwmndgplus 18100	. . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧))
9		0un 4346	. . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧
10	8, 9	syl6eq 2872	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧)
11	2, 7	pwmndgplus 18100	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅))
12	11	ancoms 461	. . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅))
13		un0 4344	. . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧
14	12, 13	syl6eq 2872	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧)
15	3, 4, 5, 6, 10, 14	ismgmid2 17878	. . 3 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ = (0g‘𝑀))
16	15	eqcomd 2827	. 2 ⊢ (∅ ∈ 𝒫 𝐴 → (0g‘𝑀) = ∅)
17	1, 16	ax-mp 5	1 ⊢ (0g‘𝑀) = ∅

Syntax hints:   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∪ cun 3934  ∅c0 4291  𝒫 cpw 4539  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  Basecbs 16483  +_gcplusg 16565  0_gc0g 16713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-0g 16715

This theorem is referenced by: (None)