Theorem r1wunlim 10806

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 10787	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ∅ ∈ (𝑅1‘𝐴))
3		elfvdm 6957	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → 𝐴 ∈ dom 𝑅1)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5		r1fnon 9836	. . . . . 6 ⊢ 𝑅1 Fn On
6	5	fndmi 6683	. . . . 5 ⊢ dom 𝑅1 = On
7	4, 6	eleqtrdi 2854	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ On)
8		eloni 6405	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
9	7, 8	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Ord 𝐴)
10		n0i 4363	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → ¬ (𝑅1‘𝐴) = ∅)
11	2, 10	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝑅1‘𝐴) = ∅)
12		fveq2 6920	. . . . . 6 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = (𝑅1‘∅))
13		r10 9837	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
14	12, 13	eqtrdi 2796	. . . . 5 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = ∅)
15	11, 14	nsyl 140	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
16		onsuc 7847	. . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
17	7, 16	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → suc 𝐴 ∈ On)
18		sucidg 6476	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
19	7, 18	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
20		r1ord 9849	. . . . . . 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 9865	. . . . . 6 ⊢ ((𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On))
23		wfelirr 9894	. . . . . 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 772	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
26	25	fveq2d 6924	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
27		r1suc 9839	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
28	27	ad2antrl 727	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
29	26, 28	eqtrd 2780	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = 𝒫 (𝑅1‘𝑥))
30		simplr 768	. . . . . . . 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 6476	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
33	32	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
34	33, 25	eleqtrrd 2847	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ 𝐴)
35		r1ord 9849	. . . . . . . . 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 10776	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1‘𝑥) ∈ (𝑅1‘𝐴))
38	29, 37	eqeltrd 2844	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
39	38	rexlimdvaa 3162	. . . . 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 984	. . . 4 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
42	15, 40, 41	sylanbrc 582	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
43		dflim3 7884	. . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
44	9, 42, 43	sylanbrc 582	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Lim 𝐴)
45		r1limwun 10805	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni)
46	44, 45	impbida 800	1 ⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝐴) ∈ WUni ↔ Lim 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931  dom cdm 5700   “ cima 5703  Ord word 6394  Oncon0 6395  Lim wlim 6396  suc csuc 6397  ‘cfv 6573  𝑅₁cr1 9831  WUnicwun 10769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-r1 9833  df-rank 9834  df-wun 10771

This theorem is referenced by: (None)