r1limwun - Metamath Proof Explorer

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Theorem r1limwun 10733

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 9773	. . 3 ⊢ Tr (𝑅₁‘𝐴)
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → Tr (𝑅₁‘𝐴))
3		limelon 6427	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4		r1fnon 9764	. . . . . . 7 ⊢ 𝑅₁ Fn On
5	4	fndmi 6652	. . . . . 6 ⊢ dom 𝑅₁ = On
6	3, 5	eleqtrrdi 2842	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅₁)
7		onssr1 9828	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅₁ → 𝐴 ⊆ (𝑅₁‘𝐴))
8	6, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅₁‘𝐴))
9		0ellim 6426	. . . . 5 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
10	9	adantl 480	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
11	8, 10	sseldd 3982	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅₁‘𝐴))
12	11	ne0d 4334	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅₁‘𝐴) ≠ ∅)
13		rankuni 9860	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
14		rankon 9792	. . . . . . . . 9 ⊢ (rank‘𝑥) ∈ On
15		eloni 6373	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
16		orduniss 6460	. . . . . . . . 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 9795	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅₁‘𝐴) → (rank‘𝑥) ∈ 𝐴)
20	19	adantl 480	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
21		onuni 7778	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → ∪ (rank‘𝑥) ∈ On)
22	14, 21	ax-mp 5	. . . . . . . 8 ⊢ ∪ (rank‘𝑥) ∈ On
23	3	adantr 479	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ On)
24		ontr2 6410	. . . . . . . 8 ⊢ ((∪ (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
25	22, 23, 24	sylancr 585	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
26	18, 20, 25	mp2and 695	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∪ (rank‘𝑥) ∈ 𝐴)
27	13, 26	eqeltrid 2835	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (rank‘∪ 𝑥) ∈ 𝐴)
28		r1elwf 9793	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅₁‘𝐴) → 𝑥 ∈ ∪ (𝑅₁ “ On))
29	28	adantl 480	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝑥 ∈ ∪ (𝑅₁ “ On))
30		uniwf 9816	. . . . . . 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 479	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ dom 𝑅₁)
33		rankr1ag 9799	. . . . . 6 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ∈ dom 𝑅₁) → (∪ 𝑥 ∈ (𝑅₁‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
34	31, 32, 33	syl2anc 582	. . . . 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 9844	. . . . . 6 ⊢ (Lim 𝐴 → (𝑥 ∈ (𝑅₁‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
37	36	adantl 480	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅₁‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
38	37	biimpa 475	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝒫 𝑥 ∈ (𝑅₁‘𝐴))
39	28	ad2antlr 723	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝑥 ∈ ∪ (𝑅₁ “ On))
40		r1elwf 9793	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑅₁‘𝐴) → 𝑦 ∈ ∪ (𝑅₁ “ On))
41	40	adantl 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝑦 ∈ ∪ (𝑅₁ “ On))
42		rankprb 9848	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝑦 ∈ ∪ (𝑅₁ “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
43	39, 41, 42	syl2anc 582	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
44		limord 6423	. . . . . . . . . 10 ⊢ (Lim 𝐴 → Ord 𝐴)
45	44	ad3antlr 727	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → Ord 𝐴)
46	20	adantr 479	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
47		rankr1ai 9795	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑅₁‘𝐴) → (rank‘𝑦) ∈ 𝐴)
48	47	adantl 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘𝑦) ∈ 𝐴)
49		ordunel 7817	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
50	45, 46, 48, 49	syl3anc 1369	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
51		limsuc 7840	. . . . . . . . 9 ⊢ (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
52	51	ad3antlr 727	. . . . . . . 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 2831	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
55		prwf 9808	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝑦 ∈ ∪ (𝑅₁ “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On))
56	39, 41, 55	syl2anc 582	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → {𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On))
57	32	adantr 479	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) ∧ 𝑦 ∈ (𝑅₁‘𝐴)) → 𝐴 ∈ dom 𝑅₁)
58		rankr1ag 9799	. . . . . . 7 ⊢ (({𝑥, 𝑦} ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ∈ dom 𝑅₁) → ({𝑥, 𝑦} ∈ (𝑅₁‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
59	56, 57, 58	syl2anc 582	. . . . . 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 3144	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴))
62	35, 38, 61	3jca 1126	. . 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 3144	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅₁‘𝐴)(∪ 𝑥 ∈ (𝑅₁‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅₁‘𝐴) ∧ ∀𝑦 ∈ (𝑅₁‘𝐴){𝑥, 𝑦} ∈ (𝑅₁‘𝐴)))
64		fvex 6903	. . 3 ⊢ (𝑅₁‘𝐴) ∈ V
65		iswun 10701	. . 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 1341	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅₁‘𝐴) ∈ WUni)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 Vcvv 3472 ∪ cun 3945 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 {cpr 4629 ∪ cuni 4907 Tr wtr 5264 dom cdm 5675 “ cima 5678 Ord word 6362 Oncon0 6363 Lim wlim 6364 suc csuc 6365 ‘cfv 6542 𝑅₁cr1 9759 rankcrnk 9760 WUnicwun 10697

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-reg 9589 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-r1 9761 df-rank 9762 df-wun 10699

This theorem is referenced by: r1wunlim 10734 wunex3 10738

W3C validator