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Theorem pwmndid 18575
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 5278 . 2 ∅ ∈ 𝒫 𝐴
2 pwmnd.b . . . . 5 (Base‘𝑀) = 𝒫 𝐴
32eqcomi 2747 . . . 4 𝒫 𝐴 = (Base‘𝑀)
4 eqid 2738 . . . 4 (0g𝑀) = (0g𝑀)
5 eqid 2738 . . . 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 18574 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = (∅ ∪ 𝑧))
9 0un 4326 . . . . 5 (∅ ∪ 𝑧) = 𝑧
108, 9eqtrdi 2794 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = 𝑧)
112, 7pwmndgplus 18574 . . . . . 6 ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
1211ancoms 459 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
13 un0 4324 . . . . 5 (𝑧 ∪ ∅) = 𝑧
1412, 13eqtrdi 2794 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = 𝑧)
153, 4, 5, 6, 10, 14ismgmid2 18352 . . 3 (∅ ∈ 𝒫 𝐴 → ∅ = (0g𝑀))
1615eqcomd 2744 . 2 (∅ ∈ 𝒫 𝐴 → (0g𝑀) = ∅)
171, 16ax-mp 5 1 (0g𝑀) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wcel 2106  cun 3885  c0 4256  𝒫 cpw 4533  cfv 6433  (class class class)co 7275  cmpo 7277  Basecbs 16912  +gcplusg 16962  0gc0g 17150
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-0g 17152
This theorem is referenced by: (None)
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