Theorem pwmndid 18846

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 5296	. 2 ⊢ ∅ ∈ 𝒫 𝐴
2		pwmnd.b	. . . . 5 ⊢ (Base‘𝑀) = 𝒫 𝐴
3	2	eqcomi 2742	. . . 4 ⊢ 𝒫 𝐴 = (Base‘𝑀)
4		eqid 2733	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
5		eqid 2733	. . . 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 18845	. . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧))
9		0un 4345	. . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧
10	8, 9	eqtrdi 2784	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧)
11	2, 7	pwmndgplus 18845	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅))
12	11	ancoms 458	. . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅))
13		un0 4343	. . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧
14	12, 13	eqtrdi 2784	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧)
15	3, 4, 5, 6, 10, 14	ismgmid2 18578	. . 3 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ = (0g‘𝑀))
16	15	eqcomd 2739	. 2 ⊢ (∅ ∈ 𝒫 𝐴 → (0g‘𝑀) = ∅)
17	1, 16	ax-mp 5	1 ⊢ (0g‘𝑀) = ∅

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∪ cun 3896  ∅c0 4282  𝒫 cpw 4549  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  Basecbs 17122  +_gcplusg 17163  0_gc0g 17345

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-0g 17347

This theorem is referenced by: (None)