Theorem pwmndid 18863

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 5311	. 2 ⊢ ∅ ∈ 𝒫 𝐴
2		pwmnd.b	. . . . 5 ⊢ (Base‘𝑀) = 𝒫 𝐴
3	2	eqcomi 2738	. . . 4 ⊢ 𝒫 𝐴 = (Base‘𝑀)
4		eqid 2729	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
5		eqid 2729	. . . 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 18862	. . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧))
9		0un 4359	. . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧
10	8, 9	eqtrdi 2780	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧)
11	2, 7	pwmndgplus 18862	. . . . . 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 4357	. . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧
14	12, 13	eqtrdi 2780	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧)
15	3, 4, 5, 6, 10, 14	ismgmid2 18595	. . 3 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ = (0g‘𝑀))
16	15	eqcomd 2735	. 2 ⊢ (∅ ∈ 𝒫 𝐴 → (0g‘𝑀) = ∅)
17	1, 16	ax-mp 5	1 ⊢ (0g‘𝑀) = ∅

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3912  ∅c0 4296  𝒫 cpw 4563  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-0g 17404

This theorem is referenced by: (None)