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Theorem pwmndid 18816
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 (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))
Assertion
Ref Expression
pwmndid (0g𝑀) = ∅
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.
StepHypRef Expression
1 0elpw 5354 . 2 ∅ ∈ 𝒫 𝐴
2 pwmnd.b . . . . 5 (Base‘𝑀) = 𝒫 𝐴
32eqcomi 2741 . . . 4 𝒫 𝐴 = (Base‘𝑀)
4 eqid 2732 . . . 4 (0g𝑀) = (0g𝑀)
5 eqid 2732 . . . 4 (+g𝑀) = (+g𝑀)
6 id 22 . . . 4 (∅ ∈ 𝒫 𝐴 → ∅ ∈ 𝒫 𝐴)
7 pwmnd.p . . . . . 6 (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))
82, 7pwmndgplus 18815 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = (∅ ∪ 𝑧))
9 0un 4392 . . . . 5 (∅ ∪ 𝑧) = 𝑧
108, 9eqtrdi 2788 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = 𝑧)
112, 7pwmndgplus 18815 . . . . . 6 ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
1211ancoms 459 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
13 un0 4390 . . . . 5 (𝑧 ∪ ∅) = 𝑧
1412, 13eqtrdi 2788 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = 𝑧)
153, 4, 5, 6, 10, 14ismgmid2 18586 . . 3 (∅ ∈ 𝒫 𝐴 → ∅ = (0g𝑀))
1615eqcomd 2738 . 2 (∅ ∈ 𝒫 𝐴 → (0g𝑀) = ∅)
171, 16ax-mp 5 1 (0g𝑀) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  cun 3946  c0 4322  𝒫 cpw 4602  cfv 6543  (class class class)co 7408  cmpo 7410  Basecbs 17143  +gcplusg 17196  0gc0g 17384
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-0g 17386
This theorem is referenced by: (None)
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