r1limwun - Metamath Proof Explorer

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Theorem r1limwun 10731

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 9771	. . 3 ⊢ Tr (𝑅₁‘𝐴)
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → Tr (𝑅₁‘𝐴))
3		limelon 6429	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4		r1fnon 9762	. . . . . . 7 ⊢ 𝑅₁ Fn On
5	4	fndmi 6654	. . . . . 6 ⊢ dom 𝑅₁ = On
6	3, 5	eleqtrrdi 2845	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅₁)
7		onssr1 9826	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅₁ → 𝐴 ⊆ (𝑅₁‘𝐴))
8	6, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅₁‘𝐴))
9		0ellim 6428	. . . . 5 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
10	9	adantl 483	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
11	8, 10	sseldd 3984	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅₁‘𝐴))
12	11	ne0d 4336	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅₁‘𝐴) ≠ ∅)
13		rankuni 9858	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
14		rankon 9790	. . . . . . . . 9 ⊢ (rank‘𝑥) ∈ On
15		eloni 6375	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
16		orduniss 6462	. . . . . . . . 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 9793	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅₁‘𝐴) → (rank‘𝑥) ∈ 𝐴)
20	19	adantl 483	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
21		onuni 7776	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → ∪ (rank‘𝑥) ∈ On)
22	14, 21	ax-mp 5	. . . . . . . 8 ⊢ ∪ (rank‘𝑥) ∈ On
23	3	adantr 482	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ On)
24		ontr2 6412	. . . . . . . 8 ⊢ ((∪ (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
25	22, 23, 24	sylancr 588	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
26	18, 20, 25	mp2and 698	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∪ (rank‘𝑥) ∈ 𝐴)
27	13, 26	eqeltrid 2838	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (rank‘∪ 𝑥) ∈ 𝐴)
28		r1elwf 9791	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅₁‘𝐴) → 𝑥 ∈ ∪ (𝑅₁ “ On))
29	28	adantl 483	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝑥 ∈ ∪ (𝑅₁ “ On))
30		uniwf 9814	. . . . . . 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 482	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ dom 𝑅₁)
33		rankr1ag 9797	. . . . . 6 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ∈ dom 𝑅₁) → (∪ 𝑥 ∈ (𝑅₁‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
34	31, 32, 33	syl2anc 585	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (∪ 𝑥 ∈ (𝑅₁‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
35	27, 34	mpbird 257	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∪ 𝑥 ∈ (𝑅₁‘𝐴))
36		r1pwcl 9842	. . . . . 6 ⊢ (Lim 𝐴 → (𝑥 ∈ (𝑅₁‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
37	36	adantl 483	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅₁‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
38	37	biimpa 478	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝒫 𝑥 ∈ (𝑅₁‘𝐴))
39	28	ad2antlr 726	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝑥 ∈ ∪ (𝑅₁ “ On))
40		r1elwf 9791	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑅₁‘𝐴) → 𝑦 ∈ ∪ (𝑅₁ “ On))
41	40	adantl 483	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝑦 ∈ ∪ (𝑅₁ “ On))
42		rankprb 9846	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝑦 ∈ ∪ (𝑅₁ “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
43	39, 41, 42	syl2anc 585	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
44		limord 6425	. . . . . . . . . 10 ⊢ (Lim 𝐴 → Ord 𝐴)
45	44	ad3antlr 730	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → Ord 𝐴)
46	20	adantr 482	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
47		rankr1ai 9793	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑅₁‘𝐴) → (rank‘𝑦) ∈ 𝐴)
48	47	adantl 483	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘𝑦) ∈ 𝐴)
49		ordunel 7815	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
50	45, 46, 48, 49	syl3anc 1372	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
51		limsuc 7838	. . . . . . . . 9 ⊢ (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
52	51	ad3antlr 730	. . . . . . . 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 2834	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
55		prwf 9806	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝑦 ∈ ∪ (𝑅₁ “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On))
56	39, 41, 55	syl2anc 585	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → {𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On))
57	32	adantr 482	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ dom 𝑅₁)
58		rankr1ag 9797	. . . . . . 7 ⊢ (({𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ∈ dom 𝑅₁) → ({𝑥, 𝑦} ∈ (𝑅₁‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
59	56, 57, 58	syl2anc 585	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → ({𝑥, 𝑦} ∈ (𝑅₁‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
60	54, 59	mpbird 257	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → {𝑥, 𝑦} ∈ (𝑅₁‘𝐴))
61	60	ralrimiva 3147	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴))
62	35, 38, 61	3jca 1129	. . 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 3147	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅₁‘𝐴)(∪ 𝑥 ∈ (𝑅₁‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅₁‘𝐴) ∧ ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴)))
64		fvex 6905	. . 3 ⊢ (𝑅₁‘𝐴) ∈ V
65		iswun 10699	. . 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 1344	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅₁‘𝐴) ∈ WUni)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∪ cun 3947 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 {cpr 4631 ∪ cuni 4909 Tr wtr 5266 dom cdm 5677 “ cima 5680 Ord word 6364 Oncon0 6365 Lim wlim 6366 suc csuc 6367 ‘cfv 6544 𝑅₁cr1 9757 rankcrnk 9758 WUnicwun 10695

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-reg 9587 ax-inf2 9636

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-r1 9759 df-rank 9760 df-wun 10697

This theorem is referenced by: r1wunlim 10732 wunex3 10736

W3C validator