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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 5232 | . 2 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | pwmnd.b | . . . . 5 ⊢ (Base‘𝑀) = 𝒫 𝐴 | |
3 | 2 | eqcomi 2748 | . . . 4 ⊢ 𝒫 𝐴 = (Base‘𝑀) |
4 | eqid 2739 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
5 | eqid 2739 | . . . 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 18228 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧)) |
9 | 0un 4291 | . . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧 | |
10 | 8, 9 | eqtrdi 2790 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧) |
11 | 2, 7 | pwmndgplus 18228 | . . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅)) |
12 | 11 | ancoms 462 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅)) |
13 | un0 4289 | . . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧 | |
14 | 12, 13 | eqtrdi 2790 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧) |
15 | 3, 4, 5, 6, 10, 14 | ismgmid2 18006 | . . 3 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ = (0g‘𝑀)) |
16 | 15 | eqcomd 2745 | . 2 ⊢ (∅ ∈ 𝒫 𝐴 → (0g‘𝑀) = ∅) |
17 | 1, 16 | ax-mp 5 | 1 ⊢ (0g‘𝑀) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∪ cun 3851 ∅c0 4221 𝒫 cpw 4498 ‘cfv 6349 (class class class)co 7182 ∈ cmpo 7184 Basecbs 16598 +gcplusg 16680 0gc0g 16828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-iota 6307 df-fun 6351 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-0g 16830 |
This theorem is referenced by: (None) |
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