Theorem r1limwun 9956

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 8999	. . 3 ⊢ Tr (𝑅1‘𝐴)
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → Tr (𝑅1‘𝐴))
3		limelon 6092	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4		r1fnon 8990	. . . . . . 7 ⊢ 𝑅1 Fn On
5		fndm 6288	. . . . . . 7 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ dom 𝑅1 = On
7	3, 6	syl6eleqr 2877	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
8		onssr1 9054	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴))
9	7, 8	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1‘𝐴))
10		0ellim 6091	. . . . 5 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
11	10	adantl 474	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
12	9, 11	sseldd 3859	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1‘𝐴))
13	12	ne0d 4187	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ≠ ∅)
14		rankuni 9086	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
15		rankon 9018	. . . . . . . . 9 ⊢ (rank‘𝑥) ∈ On
16		eloni 6039	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
17		orduniss 6123	. . . . . . . . 9 ⊢ (Ord (rank‘𝑥) → ∪ (rank‘𝑥) ⊆ (rank‘𝑥))
18	15, 16, 17	mp2b 10	. . . . . . . 8 ⊢ ∪ (rank‘𝑥) ⊆ (rank‘𝑥)
19	18	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ (rank‘𝑥) ⊆ (rank‘𝑥))
20		rankr1ai 9021	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → (rank‘𝑥) ∈ 𝐴)
21	20	adantl 474	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
22		onuni 7324	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → ∪ (rank‘𝑥) ∈ On)
23	15, 22	ax-mp 5	. . . . . . . 8 ⊢ ∪ (rank‘𝑥) ∈ On
24	3	adantr 473	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝐴 ∈ On)
25		ontr2 6076	. . . . . . . 8 ⊢ ((∪ (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
26	23, 24, 25	sylancr 578	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
27	19, 21, 26	mp2and 686	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ (rank‘𝑥) ∈ 𝐴)
28	14, 27	syl5eqel 2870	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘∪ 𝑥) ∈ 𝐴)
29		r1elwf 9019	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
30	29	adantl 474	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
31		uniwf 9042	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
32	30, 31	sylib 210	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
33	7	adantr 473	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
34		rankr1ag 9025	. . . . . 6 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (∪ 𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
35	32, 33, 34	syl2anc 576	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (∪ 𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
36	28, 35	mpbird 249	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ 𝑥 ∈ (𝑅1‘𝐴))
37		r1pwcl 9070	. . . . . 6 ⊢ (Lim 𝐴 → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
38	37	adantl 474	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
39	38	biimpa 469	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝒫 𝑥 ∈ (𝑅1‘𝐴))
40	29	ad2antlr 714	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
41		r1elwf 9019	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑅1‘𝐴) → 𝑦 ∈ ∪ (𝑅1 “ On))
42	41	adantl 474	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝑦 ∈ ∪ (𝑅1 “ On))
43		rankprb 9074	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
44	40, 42, 43	syl2anc 576	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
45		limord 6088	. . . . . . . . . 10 ⊢ (Lim 𝐴 → Ord 𝐴)
46	45	ad3antlr 718	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → Ord 𝐴)
47	21	adantr 473	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
48		rankr1ai 9021	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑅1‘𝐴) → (rank‘𝑦) ∈ 𝐴)
49	48	adantl 474	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘𝑦) ∈ 𝐴)
50		ordunel 7358	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
51	46, 47, 49, 50	syl3anc 1351	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
52		limsuc 7380	. . . . . . . . 9 ⊢ (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
53	52	ad3antlr 718	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
54	51, 53	mpbid 224	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
55	44, 54	eqeltrd 2866	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
56		prwf 9034	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
57	40, 42, 56	syl2anc 576	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
58	33	adantr 473	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
59		rankr1ag 9025	. . . . . . 7 ⊢ (({𝑥, 𝑦} ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
60	57, 58, 59	syl2anc 576	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → ({𝑥, 𝑦} ∈ (𝑅1‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
61	55, 60	mpbird 249	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → {𝑥, 𝑦} ∈ (𝑅1‘𝐴))
62	61	ralrimiva 3132	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴))
63	36, 39, 62	3jca 1108	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (∪ 𝑥 ∈ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴) ∧ ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴)))
64	63	ralrimiva 3132	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1‘𝐴)(∪ 𝑥 ∈ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴) ∧ ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴)))
65		fvex 6512	. . 3 ⊢ (𝑅1‘𝐴) ∈ V
66		iswun 9924	. . 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)))))
67	65, 66	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))))
68	2, 13, 64, 67	syl3anbrc 1323	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2967  ∀wral 3088  Vcvv 3415   ∪ cun 3827   ⊆ wss 3829  ∅c0 4178  𝒫 cpw 4422  {cpr 4443  ∪ cuni 4712  Tr wtr 5030  dom cdm 5407   “ cima 5410  Ord word 6028  Oncon0 6029  Lim wlim 6030  suc csuc 6031   Fn wfn 6183  ‘cfv 6188  𝑅₁cr1 8985  rankcrnk 8986  WUnicwun 9920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-reg 8851  ax-inf2 8898

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-r1 8987  df-rank 8988  df-wun 9922

This theorem is referenced by:  r1wunlim  9957  wunex3  9961