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Theorem gruwun 10844

Description: A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)

**Assertion**
Ref	Expression
gruwun	⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)

Proof of Theorem gruwun Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grutr 10824	. . 3 ⊢ (𝑈 ∈ Univ → Tr 𝑈)
2	1	adantr 479	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → Tr 𝑈)
3		simpr 483	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ≠ ∅)
4		gruuni 10831	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈)
5	4	adantlr 713	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈)
6		grupw 10826	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈)
7	6	adantlr 713	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈)
8		grupr 10828	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ∈ 𝑈)
9	8	ad4ant134 1171	. . . . 5 ⊢ ((((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ∈ 𝑈)
10	9	ralrimiva 3143	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)
11	5, 7, 10	3jca 1125	. . 3 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
12	11	ralrimiva 3143	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
13		iswun 10735	. . 3 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
14	13	adantr 479	. 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 1339	1 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∅c0 4326 𝒫 cpw 4606 {cpr 4634 ∪ cuni 4912 Tr wtr 5269 WUnicwun 10731 Univcgru 10821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853 df-wun 10733 df-gru 10822

This theorem is referenced by: (None)

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