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Mirrors > Home > MPE Home > Th. List > pwmndid | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
pwmnd.b | ⊢ (Base‘𝑀) = 𝒫 𝐴 |
pwmnd.p | ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦)) |
Ref | Expression |
---|---|
pwmndid | ⊢ (0g‘𝑀) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 5360 | . 2 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | pwmnd.b | . . . . 5 ⊢ (Base‘𝑀) = 𝒫 𝐴 | |
3 | 2 | eqcomi 2737 | . . . 4 ⊢ 𝒫 𝐴 = (Base‘𝑀) |
4 | eqid 2728 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
5 | eqid 2728 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
6 | id 22 | . . . 4 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ ∈ 𝒫 𝐴) | |
7 | pwmnd.p | . . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦)) | |
8 | 2, 7 | pwmndgplus 18894 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧)) |
9 | 0un 4396 | . . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧 | |
10 | 8, 9 | eqtrdi 2784 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧) |
11 | 2, 7 | pwmndgplus 18894 | . . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅)) |
12 | 11 | ancoms 457 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅)) |
13 | un0 4394 | . . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧 | |
14 | 12, 13 | eqtrdi 2784 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧) |
15 | 3, 4, 5, 6, 10, 14 | ismgmid2 18635 | . . 3 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ = (0g‘𝑀)) |
16 | 15 | eqcomd 2734 | . 2 ⊢ (∅ ∈ 𝒫 𝐴 → (0g‘𝑀) = ∅) |
17 | 1, 16 | ax-mp 5 | 1 ⊢ (0g‘𝑀) = ∅ |
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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