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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 5362 | . 2 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | pwmnd.b | . . . . 5 ⊢ (Base‘𝑀) = 𝒫 𝐴 | |
3 | 2 | eqcomi 2744 | . . . 4 ⊢ 𝒫 𝐴 = (Base‘𝑀) |
4 | eqid 2735 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
5 | eqid 2735 | . . . 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 18961 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧)) |
9 | 0un 4402 | . . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧 | |
10 | 8, 9 | eqtrdi 2791 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧) |
11 | 2, 7 | pwmndgplus 18961 | . . . . . 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 4400 | . . . . 5 ⊢ (𝑧 ∪ ∅) = 𝑧 | |
14 | 12, 13 | eqtrdi 2791 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = 𝑧) |
15 | 3, 4, 5, 6, 10, 14 | ismgmid2 18694 | . . 3 ⊢ (∅ ∈ 𝒫 𝐴 → ∅ = (0g‘𝑀)) |
16 | 15 | eqcomd 2741 | . 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 1537 ∈ wcel 2106 ∪ cun 3961 ∅c0 4339 𝒫 cpw 4605 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 +gcplusg 17298 0gc0g 17486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-0g 17488 |
This theorem is referenced by: (None) |
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