Theorem r1limwun 10659

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 9700	. . 3 ⊢ Tr (𝑅1‘𝐴)
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → Tr (𝑅1‘𝐴))
3		limelon 6390	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4		r1fnon 9691	. . . . . . 7 ⊢ 𝑅1 Fn On
5	4	fndmi 6604	. . . . . 6 ⊢ dom 𝑅1 = On
6	3, 5	eleqtrrdi 2848	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
7		onssr1 9755	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴))
8	6, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1‘𝐴))
9		0ellim 6389	. . . . 5 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
10	9	adantl 481	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
11	8, 10	sseldd 3936	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1‘𝐴))
12	11	ne0d 4296	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ≠ ∅)
13		rankuni 9787	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
14		rankon 9719	. . . . . . . . 9 ⊢ (rank‘𝑥) ∈ On
15		eloni 6335	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
16		orduniss 6424	. . . . . . . . 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 9722	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → (rank‘𝑥) ∈ 𝐴)
20	19	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘𝑥) ∈ 𝐴)
21		onuni 7743	. . . . . . . . 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 6373	. . . . . . . 8 ⊢ ((∪ (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
25	22, 23, 24	sylancr 588	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ((∪ (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → ∪ (rank‘𝑥) ∈ 𝐴))
26	18, 20, 25	mp2and 700	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ (rank‘𝑥) ∈ 𝐴)
27	13, 26	eqeltrid 2841	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (rank‘∪ 𝑥) ∈ 𝐴)
28		r1elwf 9720	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
29	28	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
30		uniwf 9743	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
31	29, 30	sylib 218	. . . . . 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 9726	. . . . . 6 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (∪ 𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
34	31, 32, 33	syl2anc 585	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (∪ 𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘∪ 𝑥) ∈ 𝐴))
35	27, 34	mpbird 257	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∪ 𝑥 ∈ (𝑅1‘𝐴))
36		r1pwcl 9771	. . . . . 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 728	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
40		r1elwf 9720	. . . . . . . . 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 9775	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
43	39, 41, 42	syl2anc 585	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
44		limord 6386	. . . . . . . . . 10 ⊢ (Lim 𝐴 → Ord 𝐴)
45	44	ad3antlr 732	. . . . . . . . 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 9722	. . . . . . . . . 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 7779	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
50	45, 46, 48, 49	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
51		limsuc 7801	. . . . . . . . 9 ⊢ (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
52	51	ad3antlr 732	. . . . . . . 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 232	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
54	43, 53	eqeltrd 2837	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
55		prwf 9735	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
56	39, 41, 55	syl2anc 585	. . . . . . 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 9726	. . . . . . 7 ⊢ (({𝑥, 𝑦} ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
59	56, 57, 58	syl2anc 585	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → ({𝑥, 𝑦} ∈ (𝑅1‘𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
60	54, 59	mpbird 257	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) ∧ 𝑦 ∈ (𝑅1‘𝐴)) → {𝑥, 𝑦} ∈ (𝑅1‘𝐴))
61	60	ralrimiva 3130	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1‘𝐴)) → ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴))
62	35, 38, 61	3jca 1129	. . 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 3130	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1‘𝐴)(∪ 𝑥 ∈ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴) ∧ ∀𝑦 ∈ (𝑅1‘𝐴){𝑥, 𝑦} ∈ (𝑅1‘𝐴)))
64		fvex 6855	. . 3 ⊢ (𝑅1‘𝐴) ∈ V
65		iswun 10627	. . 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 1345	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {cpr 4584  ∪ cuni 4865  Tr wtr 5207  dom cdm 5632   “ cima 5635  Ord word 6324  Oncon0 6325  Lim wlim 6326  suc csuc 6327  ‘cfv 6500  𝑅₁cr1 9686  rankcrnk 9687  WUnicwun 10623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-reg 9509  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-r1 9688  df-rank 9689  df-wun 10625

This theorem is referenced by:  r1wunlim  10660  wunex3  10664