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Theorem r1wunlim 9597
Description: The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
r1wunlim (𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))

Proof of Theorem r1wunlim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (𝑅1𝐴) ∈ WUni)
21wun0 9578 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ∅ ∈ (𝑅1𝐴))
3 elfvdm 6258 . . . . . 6 (∅ ∈ (𝑅1𝐴) → 𝐴 ∈ dom 𝑅1)
42, 3syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5 r1fnon 8668 . . . . . 6 𝑅1 Fn On
6 fndm 6028 . . . . . 6 (𝑅1 Fn On → dom 𝑅1 = On)
75, 6ax-mp 5 . . . . 5 dom 𝑅1 = On
84, 7syl6eleq 2740 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ On)
9 eloni 5771 . . . 4 (𝐴 ∈ On → Ord 𝐴)
108, 9syl 17 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Ord 𝐴)
11 n0i 3953 . . . . . 6 (∅ ∈ (𝑅1𝐴) → ¬ (𝑅1𝐴) = ∅)
122, 11syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) = ∅)
13 fveq2 6229 . . . . . 6 (𝐴 = ∅ → (𝑅1𝐴) = (𝑅1‘∅))
14 r10 8669 . . . . . 6 (𝑅1‘∅) = ∅
1513, 14syl6eq 2701 . . . . 5 (𝐴 = ∅ → (𝑅1𝐴) = ∅)
1612, 15nsyl 135 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
17 suceloni 7055 . . . . . . . 8 (𝐴 ∈ On → suc 𝐴 ∈ On)
188, 17syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → suc 𝐴 ∈ On)
19 sucidg 5841 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
208, 19syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
21 r1ord 8681 . . . . . . 7 (suc 𝐴 ∈ On → (𝐴 ∈ suc 𝐴 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴)))
2218, 20, 21sylc 65 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
23 r1elwf 8697 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
24 wfelirr 8726 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1 “ On) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
2522, 23, 243syl 18 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
26 simprr 811 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
2726fveq2d 6233 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = (𝑅1‘suc 𝑥))
28 r1suc 8671 . . . . . . . . 9 (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2928ad2antrl 764 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
3027, 29eqtrd 2685 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = 𝒫 (𝑅1𝑥))
31 simplr 807 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ WUni)
328adantr 480 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 ∈ On)
33 sucidg 5841 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
3433ad2antrl 764 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
3534, 26eleqtrrd 2733 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥𝐴)
36 r1ord 8681 . . . . . . . . 9 (𝐴 ∈ On → (𝑥𝐴 → (𝑅1𝑥) ∈ (𝑅1𝐴)))
3732, 35, 36sylc 65 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝑥) ∈ (𝑅1𝐴))
3831, 37wunpw 9567 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1𝑥) ∈ (𝑅1𝐴))
3930, 38eqeltrd 2730 . . . . . 6 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ (𝑅1𝐴))
4039rexlimdvaa 3061 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1𝐴) ∈ (𝑅1𝐴)))
4125, 40mtod 189 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
42 ioran 510 . . . 4 (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
4316, 41, 42sylanbrc 699 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
44 dflim3 7089 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
4510, 43, 44sylanbrc 699 . 2 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Lim 𝐴)
46 r1limwun 9596 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
4745, 46impbida 895 1 (𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wrex 2942  c0 3948  𝒫 cpw 4191   cuni 4468  dom cdm 5143  cima 5146  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763   Fn wfn 5921  cfv 5926  𝑅1cr1 8663  WUnicwun 9560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666  df-wun 9562
This theorem is referenced by: (None)
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