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Mirrors > Home > MPE Home > Th. List > gruwun | Structured version Visualization version GIF version |
Description: A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
gruwun | ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grutr 10215 | . . 3 ⊢ (𝑈 ∈ Univ → Tr 𝑈) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → Tr 𝑈) |
3 | simpr 487 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ≠ ∅) | |
4 | gruuni 10222 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈) | |
5 | 4 | adantlr 713 | . . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈) |
6 | grupw 10217 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈) | |
7 | 6 | adantlr 713 | . . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈) |
8 | grupr 10219 | . . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ∈ 𝑈) | |
9 | 8 | ad4ant134 1170 | . . . . 5 ⊢ ((((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ∈ 𝑈) |
10 | 9 | ralrimiva 3182 | . . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) |
11 | 5, 7, 10 | 3jca 1124 | . . 3 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
12 | 11 | ralrimiva 3182 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
13 | iswun 10126 | . . 3 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
14 | 13 | adantr 483 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) |
15 | 2, 3, 12, 14 | mpbir3and 1338 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∅c0 4291 𝒫 cpw 4539 {cpr 4569 ∪ cuni 4838 Tr wtr 5172 WUnicwun 10122 Univcgru 10212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-wun 10124 df-gru 10213 |
This theorem is referenced by: (None) |
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