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Theorem pwmndid 18895
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 5360 . 2 ∅ ∈ 𝒫 𝐴
2 pwmnd.b . . . . 5 (Base‘𝑀) = 𝒫 𝐴
32eqcomi 2737 . . . 4 𝒫 𝐴 = (Base‘𝑀)
4 eqid 2728 . . . 4 (0g𝑀) = (0g𝑀)
5 eqid 2728 . . . 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 18894 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = (∅ ∪ 𝑧))
9 0un 4396 . . . . 5 (∅ ∪ 𝑧) = 𝑧
108, 9eqtrdi 2784 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑧) = 𝑧)
112, 7pwmndgplus 18894 . . . . . 6 ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
1211ancoms 457 . . . . 5 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = (𝑧 ∪ ∅))
13 un0 4394 . . . . 5 (𝑧 ∪ ∅) = 𝑧
1412, 13eqtrdi 2784 . . . 4 ((∅ ∈ 𝒫 𝐴𝑧 ∈ 𝒫 𝐴) → (𝑧(+g𝑀)∅) = 𝑧)
153, 4, 5, 6, 10, 14ismgmid2 18635 . . 3 (∅ ∈ 𝒫 𝐴 → ∅ = (0g𝑀))
1615eqcomd 2734 . 2 (∅ ∈ 𝒫 𝐴 → (0g𝑀) = ∅)
171, 16ax-mp 5 1 (0g𝑀) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  cun 3947  c0 4326  𝒫 cpw 4606  cfv 6553  (class class class)co 7426  cmpo 7428  Basecbs 17187  +gcplusg 17240  0gc0g 17428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-0g 17430
This theorem is referenced by: (None)
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