Theorem r1wunlim 10777

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 484	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → (𝑅1‘𝐴) ∈ WUni)
2	1	wun0 10758	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ∅ ∈ (𝑅1‘𝐴))
3		elfvdm 6943	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → 𝐴 ∈ dom 𝑅1)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5		r1fnon 9807	. . . . . 6 ⊢ 𝑅1 Fn On
6	5	fndmi 6672	. . . . 5 ⊢ dom 𝑅1 = On
7	4, 6	eleqtrdi 2851	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ On)
8		eloni 6394	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
9	7, 8	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Ord 𝐴)
10		n0i 4340	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → ¬ (𝑅1‘𝐴) = ∅)
11	2, 10	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝑅1‘𝐴) = ∅)
12		fveq2 6906	. . . . . 6 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = (𝑅1‘∅))
13		r10 9808	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
14	12, 13	eqtrdi 2793	. . . . 5 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = ∅)
15	11, 14	nsyl 140	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
16		onsuc 7831	. . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
17	7, 16	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → suc 𝐴 ∈ On)
18		sucidg 6465	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
19	7, 18	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
20		r1ord 9820	. . . . . . 7 ⊢ (suc 𝐴 ∈ On → (𝐴 ∈ suc 𝐴 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)))
21	17, 19, 20	sylc 65	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴))
22		r1elwf 9836	. . . . . 6 ⊢ ((𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On))
23		wfelirr 9865	. . . . . 6 ⊢ ((𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On) → ¬ (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
24	21, 22, 23	3syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
25		simprr 773	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
26	25	fveq2d 6910	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
27		r1suc 9810	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
28	27	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
29	26, 28	eqtrd 2777	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = 𝒫 (𝑅1‘𝑥))
30		simplr 769	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ WUni)
31	7	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 ∈ On)
32		sucidg 6465	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
33	32	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
34	33, 25	eleqtrrd 2844	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ 𝐴)
35		r1ord 9820	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → (𝑅1‘𝑥) ∈ (𝑅1‘𝐴)))
36	31, 34, 35	sylc 65	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝑥) ∈ (𝑅1‘𝐴))
37	30, 36	wunpw 10747	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1‘𝑥) ∈ (𝑅1‘𝐴))
38	29, 37	eqeltrd 2841	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
39	38	rexlimdvaa 3156	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐴)))
40	24, 39	mtod 198	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
41		ioran 986	. . . 4 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
42	15, 40, 41	sylanbrc 583	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
43		dflim3 7868	. . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
44	9, 42, 43	sylanbrc 583	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Lim 𝐴)
45		r1limwun 10776	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni)
46	44, 45	impbida 801	1 ⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝐴) ∈ WUni ↔ Lim 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  ∅c0 4333  𝒫 cpw 4600  ∪ cuni 4907  dom cdm 5685   “ cima 5688  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  ‘cfv 6561  𝑅₁cr1 9802  WUnicwun 10740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805  df-wun 10742

This theorem is referenced by: (None)