Theorem r1limwun 10722

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 9749	. . 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 9740	. . . . . . 7 ⊢ 𝑅1 Fn On
5	4	fndmi 6641	. . . . . 6 ⊢ dom 𝑅1 = On
6	3, 5	eleqtrrdi 2874	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
7		onssr1 9804	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴))
8	6, 7	syl 18	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1‘𝐴))
9		0ellim 6427	. . . . 5 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
10	9	adantl 486	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
11	8, 10	sseldd 3939	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1‘𝐴))
12	11	ne0d 4296	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ≠ ∅)
13		rankuni 9836	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
14		rankon 9768	. . . . . . . . 9 ⊢ (rank‘𝑥) ∈ On
15		eloni 6372	. . . . . . . . 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 9771	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → (rank‘𝑥) ∈ 𝐴)
20	19	adantl 486	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
21		onuni 7788	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → ∪ (rank‘𝑥) ∈ On)
22	14, 21	ax-mp 5	. . . . . . . 8 ⊢ ∪ (rank‘𝑥) ∈ On
23	3	adantr 485	. . . . . . . 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 598	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
26	18, 20, 25	mp2and 711	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ (rank‘𝑥) ∈ 𝐴)
27	13, 26	eqeltrid 2867	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘∪ 𝑥) ∈ 𝐴)
28		r1elwf 9769	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
29	28	adantl 486	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
30		uniwf 9792	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
31	29, 30	sylib 221	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
32	6	adantr 485	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
33		rankr1ag 9775	. . . . . 6 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (∪ 𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
34	31, 32, 33	syl2anc 595	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (∪ 𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
35	27, 34	mpbird 260	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ 𝑥 ∈ (𝑅1‘𝐴))
36		r1pwcl 9820	. . . . . 6 ⊢ (Lim 𝐴 → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
37	36	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
38	37	biimpa 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝒫 𝑥 ∈ (𝑅1‘𝐴))
39	28	ad2antlr 739	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
40		r1elwf 9769	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑅1‘𝐴) → 𝑦 ∈ ∪ (𝑅1 “ On))
41	40	adantl 486	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝑦 ∈ ∪ (𝑅1 “ On))
42		rankprb 9824	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
43	39, 41, 42	syl2anc 595	. . . . . . 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 743	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → Ord 𝐴)
46	20	adantr 485	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
47		rankr1ai 9771	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑅1‘𝐴) → (rank‘𝑦) ∈ 𝐴)
48	47	adantl 486	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘𝑦) ∈ 𝐴)
49		ordunel 7824	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
50	45, 46, 48, 49	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
51		limsuc 7846	. . . . . . . . 9 ⊢ (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
52	51	ad3antlr 743	. . . . . . . 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 235	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
54	43, 53	eqeltrd 2863	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
55		prwf 9784	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
56	39, 41, 55	syl2anc 595	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
57	32	adantr 485	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
58		rankr1ag 9775	. . . . . . 7 ⊢ (({𝑥, 𝑦} ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
59	56, 57, 58	syl2anc 595	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → ({𝑥, 𝑦} ∈ (𝑅1‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
60	54, 59	mpbird 260	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → {𝑥, 𝑦} ∈ (𝑅1‘𝐴))
61	60	ralrimiva 3157	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴))
62	35, 38, 61	3jca 1146	. . 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 3157	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1‘𝐴)(∪ 𝑥 ∈ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴) ∧ ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴)))
64		fvex 6896	. . 3 ⊢ (𝑅1‘𝐴) ∈ V
65		iswun 10690	. . 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 1362	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {cpr 4592  ∪ cuni 4873  Tr wtr 5219  dom cdm 5663   “ cima 5666  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364  ‘cfv 6538  𝑅₁cr1 9735  rankcrnk 9736  WUnicwun 10686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-reg 9555  ax-inf2 9611

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-r1 9737  df-rank 9738  df-wun 10688

This theorem is referenced by:  r1wunlim  10723  wunex3  10727