Theorem pwmndid 18898

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 5293	. 2 ⊢ ∅ ∈ 𝒫 𝐴
2		pwmnd.b	. . . . 5 ⊢ (Base‘𝑀) = 𝒫 𝐴
3	2	eqcomi 2746	. . . 4 ⊢ 𝒫 𝐴 = (Base‘𝑀)
4		eqid 2737	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
5		eqid 2737	. . . 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 18897	. . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧))
9		0un 4337	. . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧
10	8, 9	eqtrdi 2788	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧)
11	2, 7	pwmndgplus 18897	. . . . . 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 4335	. . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧
14	12, 13	eqtrdi 2788	. . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧)
15	3, 4, 5, 6, 10, 14	ismgmid2 18627	. . 3 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ = (0g‘𝑀))
16	15	eqcomd 2743	. 2 ⊢ (∅ ∈ 𝒫 𝐴 → (0g‘𝑀) = ∅)
17	1, 16	ax-mp 5	1 ⊢ (0g‘𝑀) = ∅

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∪ cun 3888  ∅c0 4274  𝒫 cpw 4542  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  Basecbs 17170  +_gcplusg 17211  0_gc0g 17393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-0g 17395

This theorem is referenced by: (None)