Theorem pwmndid 18813

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 5353	. 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 18812	. . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧))
9		0un 4391	. . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧
10	8, 9	eqtrdi 2789	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧)
11	2, 7	pwmndgplus 18812	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅))
12	11	ancoms 460	. . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅))
13		un0 4389	. . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧
14	12, 13	eqtrdi 2789	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧)
15	3, 4, 5, 6, 10, 14	ismgmid2 18583	. . 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 397   = wceq 1542   ∈ wcel 2107   ∪ cun 3945  ∅c0 4321  𝒫 cpw 4601  ‘cfv 6540  (class class class)co 7404   ∈ cmpo 7406  Basecbs 17140  +_gcplusg 17193  0_gc0g 17381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-0g 17383

This theorem is referenced by: (None)