r1limwun - Metamath Proof Explorer

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Theorem r1limwun 10730

Description: Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)

**Assertion**
Ref	Expression
r1limwun	⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅₁‘𝐴) ∈ WUni)

Proof of Theorem r1limwun Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1tr 9770	. . 3 ⊢ Tr (𝑅₁‘𝐴)
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → Tr (𝑅₁‘𝐴))
3		limelon 6428	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4		r1fnon 9761	. . . . . . 7 ⊢ 𝑅₁ Fn On
5	4	fndmi 6653	. . . . . 6 ⊢ dom 𝑅₁ = On
6	3, 5	eleqtrrdi 2844	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅₁)
7		onssr1 9825	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅₁ → 𝐴 ⊆ (𝑅₁‘𝐴))
8	6, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅₁‘𝐴))
9		0ellim 6427	. . . . 5 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
10	9	adantl 482	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
11	8, 10	sseldd 3983	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅₁‘𝐴))
12	11	ne0d 4335	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅₁‘𝐴) ≠ ∅)
13		rankuni 9857	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
14		rankon 9789	. . . . . . . . 9 ⊢ (rank‘𝑥) ∈ On
15		eloni 6374	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
16		orduniss 6461	. . . . . . . . 9 ⊢ (Ord (rank‘𝑥) → ∪ (rank‘𝑥) ⊆ (rank‘𝑥))
17	14, 15, 16	mp2b 10	. . . . . . . 8 ⊢ ∪ (rank‘𝑥) ⊆ (rank‘𝑥)
18	17	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∪ (rank‘𝑥) ⊆ (rank‘𝑥))
19		rankr1ai 9792	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅₁‘𝐴) → (rank‘𝑥) ∈ 𝐴)
20	19	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
21		onuni 7775	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → ∪ (rank‘𝑥) ∈ On)
22	14, 21	ax-mp 5	. . . . . . . 8 ⊢ ∪ (rank‘𝑥) ∈ On
23	3	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ On)
24		ontr2 6411	. . . . . . . 8 ⊢ ((∪ (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
25	22, 23, 24	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
26	18, 20, 25	mp2and 697	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∪ (rank‘𝑥) ∈ 𝐴)
27	13, 26	eqeltrid 2837	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (rank‘∪ 𝑥) ∈ 𝐴)
28		r1elwf 9790	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅₁‘𝐴) → 𝑥 ∈ ∪ (𝑅₁ “ On))
29	28	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝑥 ∈ ∪ (𝑅₁ “ On))
30		uniwf 9813	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅₁ “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅₁ “ On))
31	29, 30	sylib 217	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∪ 𝑥 ∈ ∪ (𝑅₁ “ On))
32	6	adantr 481	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ dom 𝑅₁)
33		rankr1ag 9796	. . . . . 6 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ∈ dom 𝑅₁) → (∪ 𝑥 ∈ (𝑅₁‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
34	31, 32, 33	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (∪ 𝑥 ∈ (𝑅₁‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
35	27, 34	mpbird 256	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∪ 𝑥 ∈ (𝑅₁‘𝐴))
36		r1pwcl 9841	. . . . . 6 ⊢ (Lim 𝐴 → (𝑥 ∈ (𝑅₁‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
37	36	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅₁‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
38	37	biimpa 477	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝒫 𝑥 ∈ (𝑅₁‘𝐴))
39	28	ad2antlr 725	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝑥 ∈ ∪ (𝑅₁ “ On))
40		r1elwf 9790	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑅₁‘𝐴) → 𝑦 ∈ ∪ (𝑅₁ “ On))
41	40	adantl 482	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝑦 ∈ ∪ (𝑅₁ “ On))
42		rankprb 9845	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝑦 ∈ ∪ (𝑅₁ “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
43	39, 41, 42	syl2anc 584	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
44		limord 6424	. . . . . . . . . 10 ⊢ (Lim 𝐴 → Ord 𝐴)
45	44	ad3antlr 729	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → Ord 𝐴)
46	20	adantr 481	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
47		rankr1ai 9792	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑅₁‘𝐴) → (rank‘𝑦) ∈ 𝐴)
48	47	adantl 482	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘𝑦) ∈ 𝐴)
49		ordunel 7814	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
50	45, 46, 48, 49	syl3anc 1371	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
51		limsuc 7837	. . . . . . . . 9 ⊢ (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
52	51	ad3antlr 729	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
53	50, 52	mpbid 231	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
54	43, 53	eqeltrd 2833	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
55		prwf 9805	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝑦 ∈ ∪ (𝑅₁ “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On))
56	39, 41, 55	syl2anc 584	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → {𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On))
57	32	adantr 481	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ dom 𝑅₁)
58		rankr1ag 9796	. . . . . . 7 ⊢ (({𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ∈ dom 𝑅₁) → ({𝑥, 𝑦} ∈ (𝑅₁‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
59	56, 57, 58	syl2anc 584	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → ({𝑥, 𝑦} ∈ (𝑅₁‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
60	54, 59	mpbird 256	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → {𝑥, 𝑦} ∈ (𝑅₁‘𝐴))
61	60	ralrimiva 3146	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴))
62	35, 38, 61	3jca 1128	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (∪ 𝑥 ∈ (𝑅₁‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅₁‘𝐴) ∧ ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴)))
63	62	ralrimiva 3146	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅₁‘𝐴)(∪ 𝑥 ∈ (𝑅₁‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅₁‘𝐴) ∧ ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴)))
64		fvex 6904	. . 3 ⊢ (𝑅₁‘𝐴) ∈ V
65		iswun 10698	. . 3 ⊢ ((𝑅₁‘𝐴) ∈ V → ((𝑅₁‘𝐴) ∈ WUni ↔ (Tr (𝑅₁‘𝐴) ∧ (𝑅₁‘𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (𝑅₁‘𝐴)(∪ 𝑥 ∈ (𝑅₁‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅₁‘𝐴) ∧ ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴)))))
66	64, 65	ax-mp 5	. 2 ⊢ ((𝑅₁‘𝐴) ∈ WUni ↔ (Tr (𝑅₁‘𝐴) ∧ (𝑅₁‘𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (𝑅₁‘𝐴)(∪ 𝑥 ∈ (𝑅₁‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅₁‘𝐴) ∧ ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴))))
67	2, 12, 63, 66	syl3anbrc 1343	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅₁‘𝐴) ∈ WUni)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 Vcvv 3474 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {cpr 4630 ∪ cuni 4908 Tr wtr 5265 dom cdm 5676 “ cima 5679 Ord word 6363 Oncon0 6364 Lim wlim 6365 suc csuc 6366 ‘cfv 6543 𝑅₁cr1 9756 rankcrnk 9757 WUnicwun 10694

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-reg 9586 ax-inf2 9635

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-r1 9758 df-rank 9759 df-wun 10696

This theorem is referenced by: r1wunlim 10731 wunex3 10735

W3C validator