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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 9494 | . . 3 ⊢ (𝑈 ∈ Univ → Tr 𝑈) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → Tr 𝑈) |
3 | simpr 476 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ≠ ∅) | |
4 | gruuni 9501 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈) | |
5 | 4 | adantlr 747 | . . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈) |
6 | grupw 9496 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈) | |
7 | 6 | adantlr 747 | . . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈) |
8 | grupr 9498 | . . . . . . 7 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ∈ 𝑈) | |
9 | 8 | 3expa 1257 | . . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ∈ 𝑈) |
10 | 9 | adantllr 751 | . . . . 5 ⊢ ((((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ∈ 𝑈) |
11 | 10 | ralrimiva 2949 | . . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) |
12 | 5, 7, 11 | 3jca 1235 | . . 3 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
13 | 12 | ralrimiva 2949 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
14 | iswun 9405 | . . 3 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
15 | 14 | adantr 480 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) |
16 | 2, 3, 13, 15 | mpbir3and 1238 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∅c0 3874 𝒫 cpw 4108 {cpr 4127 ∪ cuni 4372 Tr wtr 4680 WUnicwun 9401 Univcgru 9491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-wun 9403 df-gru 9492 |
This theorem is referenced by: (None) |
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