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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 (+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 5353 . 2 ∅ ∈ 𝒫 𝐴
2 pwmnd.b . . . . 5 (Base‘𝑀) = 𝒫 𝐴
32eqcomi 2742 . . . 4 𝒫 𝐴 = (Base‘𝑀)
4 eqid 2733 . . . 4 (0g𝑀) = (0g𝑀)
5 eqid 2733 . . . 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 18812 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = (∅ ∪ 𝑧))
9 0un 4391 . . . . 5 (∅ ∪ 𝑧) = 𝑧
108, 9eqtrdi 2789 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = 𝑧)
112, 7pwmndgplus 18812 . . . . . 6 ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
1211ancoms 460 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
13 un0 4389 . . . . 5 (𝑧 ∪ ∅) = 𝑧
1412, 13eqtrdi 2789 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = 𝑧)
153, 4, 5, 6, 10, 14ismgmid2 18583 . . 3 (∅ ∈ 𝒫 𝐴 → ∅ = (0g𝑀))
1615eqcomd 2739 . 2 (∅ ∈ 𝒫 𝐴 → (0g𝑀) = ∅)
171, 16ax-mp 5 1 (0g𝑀) = ∅
Colors of variables: wff setvar class
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  0gc0g 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)
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