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Mirrors > Home > MPE Home > Th. List > grupw | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
grupw | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elgrug 10217 | . . . . 5 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈)))) | |
2 | 1 | ibi 269 | . . . 4 ⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈))) |
3 | 2 | simprd 498 | . . 3 ⊢ (𝑈 ∈ Univ → ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈)) |
4 | simp1 1132 | . . . 4 ⊢ ((𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈) → 𝒫 𝑦 ∈ 𝑈) | |
5 | 4 | ralimi 3163 | . . 3 ⊢ (∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 𝒫 𝑦 ∈ 𝑈) |
6 | pweq 4558 | . . . . 5 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴) | |
7 | 6 | eleq1d 2900 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝒫 𝑦 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈)) |
8 | 7 | rspccv 3623 | . . 3 ⊢ (∀𝑦 ∈ 𝑈 𝒫 𝑦 ∈ 𝑈 → (𝐴 ∈ 𝑈 → 𝒫 𝐴 ∈ 𝑈)) |
9 | 3, 5, 8 | 3syl 18 | . 2 ⊢ (𝑈 ∈ Univ → (𝐴 ∈ 𝑈 → 𝒫 𝐴 ∈ 𝑈)) |
10 | 9 | imp 409 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 𝒫 cpw 4542 {cpr 4572 ∪ cuni 4841 Tr wtr 5175 ran crn 5559 (class class class)co 7159 ↑m cmap 8409 Univcgru 10215 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-tr 5176 df-iota 6317 df-fv 6366 df-ov 7162 df-gru 10216 |
This theorem is referenced by: gruss 10221 grurn 10226 gruxp 10232 grumap 10233 gruwun 10238 intgru 10239 gruina 10243 grur1a 10244 grur1cld 40574 grumnudlem 40627 |
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