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Theorem pwmndid 18949
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 5356 . 2 ∅ ∈ 𝒫 𝐴
2 pwmnd.b . . . . 5 (Base‘𝑀) = 𝒫 𝐴
32eqcomi 2746 . . . 4 𝒫 𝐴 = (Base‘𝑀)
4 eqid 2737 . . . 4 (0g𝑀) = (0g𝑀)
5 eqid 2737 . . . 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 18948 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = (∅ ∪ 𝑧))
9 0un 4396 . . . . 5 (∅ ∪ 𝑧) = 𝑧
108, 9eqtrdi 2793 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = 𝑧)
112, 7pwmndgplus 18948 . . . . . 6 ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
1211ancoms 458 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
13 un0 4394 . . . . 5 (𝑧 ∪ ∅) = 𝑧
1412, 13eqtrdi 2793 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = 𝑧)
153, 4, 5, 6, 10, 14ismgmid2 18681 . . 3 (∅ ∈ 𝒫 𝐴 → ∅ = (0g𝑀))
1615eqcomd 2743 . 2 (∅ ∈ 𝒫 𝐴 → (0g𝑀) = ∅)
171, 16ax-mp 5 1 (0g𝑀) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  cun 3949  c0 4333  𝒫 cpw 4600  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  0gc0g 17484
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-0g 17486
This theorem is referenced by: (None)
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