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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 10208 | . . . . 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 3160 | . . 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 4542 | . . . . 5 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴) | |
7 | 6 | eleq1d 2897 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝒫 𝑦 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈)) |
8 | 7 | rspccv 3620 | . . 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 1533 ∈ wcel 2110 ∀wral 3138 𝒫 cpw 4539 {cpr 4563 ∪ cuni 4832 Tr wtr 5165 ran crn 5551 (class class class)co 7150 ↑m cmap 8400 Univcgru 10206 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-tr 5166 df-iota 6309 df-fv 6358 df-ov 7153 df-gru 10207 |
This theorem is referenced by: gruss 10212 grurn 10217 gruxp 10223 grumap 10224 gruwun 10229 intgru 10230 gruina 10234 grur1a 10235 grur1cld 40561 grumnudlem 40614 |
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