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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 5312 | . 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 18746 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = (∅ ∪ 𝑧)) |
9 | 0un 4353 | . . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧 | |
10 | 8, 9 | eqtrdi 2793 | . . . 4 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑧) = 𝑧) |
11 | 2, 7 | pwmndgplus 18746 | . . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅)) |
12 | 11 | ancoms 460 | . . . . 5 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑧 ∈ 𝒫 𝐴) → (𝑧(+g‘𝑀)∅) = (𝑧 ∪ ∅)) |
13 | un0 4351 | . . . . 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 18524 | . . 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 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 ∅c0 4283 𝒫 cpw 4561 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 Basecbs 17084 +gcplusg 17134 0gc0g 17322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-0g 17324 |
This theorem is referenced by: (None) |
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