Theorem r1wunlim 10835

Description: The stages of the cumulative hierarchy of sets that are weak universes are exactly the stages at 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 490	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → (𝑅1‘𝐴) ∈ WUni)
2	1	wun0 10816	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ∅ ∈ (𝑅1‘𝐴))
3		elfvdm 6930	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → 𝐴 ∈ dom 𝑅1)
4	2, 3	syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5		r1fnon 9786	. . . . . 6 ⊢ 𝑅1 Fn On
6	5	fndmi 6654	. . . . 5 ⊢ dom 𝑅1 = On
7	4, 6	eleqtrdi 2874	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ On)
8		eloni 6379	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
9	7, 8	syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Ord 𝐴)
10		n0i 4289	. . . . . 6 ⊢ (∅ ∈ (𝑅1‘𝐴) → ¬ (𝑅1‘𝐴) = ∅)
11	2, 10	syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ (𝑅1‘𝐴) = ∅)
12		fveq2 6896	. . . . . 6 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = (𝑅1‘∅))
13		r10 9788	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
14	12, 13	eqtrdi 2814	. . . . 5 ⊢ (𝐴 = ∅ → (𝑅1‘𝐴) = ∅)
15	11, 14	nsyl 141	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
16		onsuc 7834	. . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
17	7, 16	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → suc 𝐴 ∈ On)
18		sucidg 6453	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
19	7, 18	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
20		r1ord 9800	. . . . . . 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 9817	. . . . . 6 ⊢ ((𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On))
23		wfelirr 9847	. . . . . 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 785	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
26	25	fveq2d 6900	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
27		r1suc 9790	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
28	27	ad2antrl 741	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
29	26, 28	eqtrd 2798	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = 𝒫 (𝑅1‘𝑥))
30		simplr 781	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ WUni)
31	7	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 ∈ On)
32		sucidg 6453	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
33	32	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
34	33, 25	eleqtrrd 2867	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ 𝐴)
35		r1ord 9800	. . . . . . . . 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 10805	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1‘𝑥) ∈ (𝑅1‘𝐴))
38	29, 37	eqeltrd 2864	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐴))
39	38	rexlimdvaa 3168	. . . . 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 595	. . 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 595	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Lim 𝐴)
45		r1limwun 10834	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni)
46	44, 45	impbida 813	1 ⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝐴) ∈ WUni ↔ Lim 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2146  ∃wrex 3090  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870  dom cdm 5655   “ cima 5658  Ord word 6368  Oncon0 6369  Lim wlim 6370  suc csuc 6371  ‘cfv 6547  𝑅₁cr1 9779  WUnicwun 10798

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-reg 9599  ax-inf2 9655

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-r1 9781  df-rank 9782  df-wun 10800

This theorem is used by: (None)