Theorem r1wunlim 10697

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 10678	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ∅ ∈ (𝑅1‘𝐴))
3		elfvdm 6898	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → 𝐴 ∈ dom 𝑅1)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5		r1fnon 9727	. . . . . 6 ⊢ 𝑅1 Fn On
6	5	fndmi 6625	. . . . 5 ⊢ dom 𝑅1 = On
7	4, 6	eleqtrdi 2839	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ On)
8		eloni 6345	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
9	7, 8	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Ord 𝐴)
10		n0i 4306	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → ¬ (𝑅1‘𝐴) = ∅)
11	2, 10	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝑅1‘𝐴) = ∅)
12		fveq2 6861	. . . . . 6 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = (𝑅1‘∅))
13		r10 9728	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
14	12, 13	eqtrdi 2781	. . . . 5 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = ∅)
15	11, 14	nsyl 140	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
16		onsuc 7790	. . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
17	7, 16	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → suc 𝐴 ∈ On)
18		sucidg 6418	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
19	7, 18	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
20		r1ord 9740	. . . . . . 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 9756	. . . . . 6 ⊢ ((𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On))
23		wfelirr 9785	. . . . . 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 6865	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
27		r1suc 9730	. . . . . . . . 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 2765	. . . . . . 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 6418	. . . . . . . . . . 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 2832	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ 𝐴)
35		r1ord 9740	. . . . . . . . 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 10667	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1‘𝑥) ∈ (𝑅1‘𝐴))
38	29, 37	eqeltrd 2829	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
39	38	rexlimdvaa 3136	. . . . 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 985	. . . 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 7826	. . 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 10696	. 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 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  ∅c0 4299  𝒫 cpw 4566  ∪ cuni 4874  dom cdm 5641   “ cima 5644  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337  ‘cfv 6514  𝑅₁cr1 9722  WUnicwun 10660

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-r1 9724  df-rank 9725  df-wun 10662

This theorem is referenced by: (None)