Theorem r1limwun 10423

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 9465	. . 3 ⊢ Tr (𝑅1‘𝐴)
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → Tr (𝑅1‘𝐴))
3		limelon 6314	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4		r1fnon 9456	. . . . . . 7 ⊢ 𝑅1 Fn On
5	4	fndmi 6521	. . . . . 6 ⊢ dom 𝑅1 = On
6	3, 5	eleqtrrdi 2850	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
7		onssr1 9520	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴))
8	6, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1‘𝐴))
9		0ellim 6313	. . . . 5 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
10	9	adantl 481	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
11	8, 10	sseldd 3918	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1‘𝐴))
12	11	ne0d 4266	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ≠ ∅)
13		rankuni 9552	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
14		rankon 9484	. . . . . . . . 9 ⊢ (rank‘𝑥) ∈ On
15		eloni 6261	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
16		orduniss 6345	. . . . . . . . 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 9487	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → (rank‘𝑥) ∈ 𝐴)
20	19	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
21		onuni 7615	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → ∪ (rank‘𝑥) ∈ On)
22	14, 21	ax-mp 5	. . . . . . . 8 ⊢ ∪ (rank‘𝑥) ∈ On
23	3	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝐴 ∈ On)
24		ontr2 6298	. . . . . . . 8 ⊢ ((∪ (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
25	22, 23, 24	sylancr 586	. . . . . . 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 2843	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘∪ 𝑥) ∈ 𝐴)
28		r1elwf 9485	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
29	28	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
30		uniwf 9508	. . . . . . 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 480	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
33		rankr1ag 9491	. . . . . 6 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (∪ 𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
34	31, 32, 33	syl2anc 583	. . . . 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 9536	. . . . . 6 ⊢ (Lim 𝐴 → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
37	36	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
38	37	biimpa 476	. . . 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 9485	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑅1‘𝐴) → 𝑦 ∈ ∪ (𝑅1 “ On))
41	40	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝑦 ∈ ∪ (𝑅1 “ On))
42		rankprb 9540	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
43	39, 41, 42	syl2anc 583	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
44		limord 6310	. . . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
47		rankr1ai 9487	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑅1‘𝐴) → (rank‘𝑦) ∈ 𝐴)
48	47	adantl 481	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘𝑦) ∈ 𝐴)
49		ordunel 7649	. . . . . . . . 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 7671	. . . . . . . . 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 2839	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
55		prwf 9500	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
56	39, 41, 55	syl2anc 583	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
57	32	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
58		rankr1ag 9491	. . . . . . 7 ⊢ (({𝑥, 𝑦} ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
59	56, 57, 58	syl2anc 583	. . . . . 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 3107	. . . 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 3107	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1‘𝐴)(∪ 𝑥 ∈ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴) ∧ ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴)))
64		fvex 6769	. . 3 ⊢ (𝑅1‘𝐴) ∈ V
65		iswun 10391	. . 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)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {cpr 4560  ∪ cuni 4836  Tr wtr 5187  dom cdm 5580   “ cima 5583  Ord word 6250  Oncon0 6251  Lim wlim 6252  suc csuc 6253  ‘cfv 6418  𝑅₁cr1 9451  rankcrnk 9452  WUnicwun 10387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453  df-rank 9454  df-wun 10389

This theorem is referenced by:  r1wunlim  10424  wunex3  10428