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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 5356 | . 2 ⊢ ∅ ∈ 𝒫 𝐴 | |
| 2 | pwmnd.b | . . . . 5 ⊢ (Base‘𝑀) = 𝒫 𝐴 | |
| 3 | 2 | eqcomi 2746 | . . . 4 ⊢ 𝒫 𝐴 = (Base‘𝑀) |
| 4 | eqid 2737 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
| 5 | eqid 2737 | . . . 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 18948 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧)) |
| 9 | 0un 4396 | . . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧 | |
| 10 | 8, 9 | eqtrdi 2793 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧) |
| 11 | 2, 7 | pwmndgplus 18948 | . . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅)) |
| 12 | 11 | ancoms 458 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅)) |
| 13 | un0 4394 | . . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧 | |
| 14 | 12, 13 | eqtrdi 2793 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧) |
| 15 | 3, 4, 5, 6, 10, 14 | ismgmid2 18681 | . . 3 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ = (0g‘𝑀)) |
| 16 | 15 | eqcomd 2743 | . 2 ⊢ (∅ ∈ 𝒫 𝐴 → (0g‘𝑀) = ∅) |
| 17 | 1, 16 | ax-mp 5 | 1 ⊢ (0g‘𝑀) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∅c0 4333 𝒫 cpw 4600 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 +gcplusg 17297 0gc0g 17484 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-0g 17486 |
| This theorem is referenced by: (None) |
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