Theorem r1limwun 10761

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 9801	. . 3 ⊢ Tr (𝑅1‘𝐴)
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → Tr (𝑅1‘𝐴))
3		limelon 6435	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4		r1fnon 9792	. . . . . . 7 ⊢ 𝑅1 Fn On
5	4	fndmi 6659	. . . . . 6 ⊢ dom 𝑅1 = On
6	3, 5	eleqtrrdi 2836	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
7		onssr1 9856	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴))
8	6, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1‘𝐴))
9		0ellim 6434	. . . . 5 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
10	9	adantl 480	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
11	8, 10	sseldd 3977	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1‘𝐴))
12	11	ne0d 4335	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ≠ ∅)
13		rankuni 9888	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
14		rankon 9820	. . . . . . . . 9 ⊢ (rank‘𝑥) ∈ On
15		eloni 6381	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
16		orduniss 6468	. . . . . . . . 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 9823	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → (rank‘𝑥) ∈ 𝐴)
20	19	adantl 480	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
21		onuni 7792	. . . . . . . . 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 6418	. . . . . . . 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 697	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ (rank‘𝑥) ∈ 𝐴)
27	13, 26	eqeltrid 2829	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘∪ 𝑥) ∈ 𝐴)
28		r1elwf 9821	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
29	28	adantl 480	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
30		uniwf 9844	. . . . . . 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 9827	. . . . . 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 9872	. . . . . 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 725	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
40		r1elwf 9821	. . . . . . . . 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 9876	. . . . . . . 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 6431	. . . . . . . . . 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 479	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
47		rankr1ai 9823	. . . . . . . . . 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 7831	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
50	45, 46, 48, 49	syl3anc 1368	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
51		limsuc 7854	. . . . . . . . 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 2825	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
55		prwf 9836	. . . . . . . 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 9827	. . . . . . 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 3135	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴))
62	35, 38, 61	3jca 1125	. . 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 3135	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1‘𝐴)(∪ 𝑥 ∈ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴) ∧ ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴)))
64		fvex 6909	. . 3 ⊢ (𝑅1‘𝐴) ∈ V
65		iswun 10729	. . 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 1340	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  Vcvv 3461   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  𝒫 cpw 4604  {cpr 4632  ∪ cuni 4909  Tr wtr 5266  dom cdm 5678   “ cima 5681  Ord word 6370  Oncon0 6371  Lim wlim 6372  suc csuc 6373  ‘cfv 6549  𝑅₁cr1 9787  rankcrnk 9788  WUnicwun 10725

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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-reg 9617  ax-inf2 9666

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-r1 9789  df-rank 9790  df-wun 10727

This theorem is referenced by:  r1wunlim  10762  wunex3  10766