Theorem r1wunlim 10722

Description: The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
r1wunlim	⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝐴) ∈ WUni ↔ Lim 𝐴))

Proof of Theorem r1wunlim

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 489	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → (𝑅1‘𝐴) ∈ WUni)
2	1	wun0 10703	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ∅ ∈ (𝑅1‘𝐴))
3		elfvdm 6916	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → 𝐴 ∈ dom 𝑅1)
4	2, 3	syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5		r1fnon 9739	. . . . . 6 ⊢ 𝑅1 Fn On
6	5	fndmi 6640	. . . . 5 ⊢ dom 𝑅1 = On
7	4, 6	eleqtrdi 2879	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ On)
8		eloni 6371	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
9	7, 8	syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Ord 𝐴)
10		n0i 4301	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → ¬ (𝑅1‘𝐴) = ∅)
11	2, 10	syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝑅1‘𝐴) = ∅)
12		fveq2 6882	. . . . . 6 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = (𝑅1‘∅))
13		r10 9740	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
14	12, 13	eqtrdi 2820	. . . . 5 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = ∅)
15	11, 14	nsyl 141	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
16		onsuc 7809	. . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
17	7, 16	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → suc 𝐴 ∈ On)
18		sucidg 6445	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
19	7, 18	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
20		r1ord 9752	. . . . . . 7 ⊢ (suc 𝐴 ∈ On → (𝐴 ∈ suc 𝐴 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)))
21	17, 19, 20	sylc 66	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴))
22		r1elwf 9768	. . . . . 6 ⊢ ((𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On))
23		wfelirr 9797	. . . . . 6 ⊢ ((𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On) → ¬ (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
24	21, 22, 23	3syl 19	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
25		simprr 784	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
26	25	fveq2d 6886	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
27		r1suc 9742	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
28	27	ad2antrl 740	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
29	26, 28	eqtrd 2804	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = 𝒫 (𝑅1‘𝑥))
30		simplr 780	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ WUni)
31	7	adantr 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 ∈ On)
32		sucidg 6445	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
33	32	ad2antrl 740	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
34	33, 25	eleqtrrd 2872	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ 𝐴)
35		r1ord 9752	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → (𝑅1‘𝑥) ∈ (𝑅1‘𝐴)))
36	31, 34, 35	sylc 66	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝑥) ∈ (𝑅1‘𝐴))
37	30, 36	wunpw 10692	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1‘𝑥) ∈ (𝑅1‘𝐴))
38	29, 37	eqeltrd 2869	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
39	38	rexlimdvaa 3173	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐴)))
40	24, 39	mtod 201	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
41		ioran 999	. . . 4 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
42	15, 40, 41	sylanbrc 594	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
43		dflim3 7843	. . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
44	9, 42, 43	sylanbrc 594	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Lim 𝐴)
45		r1limwun 10721	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni)
46	44, 45	impbida 812	1 ⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝐴) ∈ WUni ↔ Lim 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876  dom cdm 5662   “ cima 5665  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  ‘cfv 6537  𝑅₁cr1 9734  WUnicwun 10685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-reg 9554  ax-inf2 9610

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-r1 9736  df-rank 9737  df-wun 10687

This theorem is referenced by: (None)