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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 (𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))

Proof of Theorem r1wunlim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (𝑅1𝐴) ∈ WUni)
21wun0 10758 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ∅ ∈ (𝑅1𝐴))
3 elfvdm 6943 . . . . . 6 (∅ ∈ (𝑅1𝐴) → 𝐴 ∈ dom 𝑅1)
42, 3syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5 r1fnon 9807 . . . . . 6 𝑅1 Fn On
65fndmi 6672 . . . . 5 dom 𝑅1 = On
74, 6eleqtrdi 2851 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ On)
8 eloni 6394 . . . 4 (𝐴 ∈ On → Ord 𝐴)
97, 8syl 17 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Ord 𝐴)
10 n0i 4340 . . . . . 6 (∅ ∈ (𝑅1𝐴) → ¬ (𝑅1𝐴) = ∅)
112, 10syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) = ∅)
12 fveq2 6906 . . . . . 6 (𝐴 = ∅ → (𝑅1𝐴) = (𝑅1‘∅))
13 r10 9808 . . . . . 6 (𝑅1‘∅) = ∅
1412, 13eqtrdi 2793 . . . . 5 (𝐴 = ∅ → (𝑅1𝐴) = ∅)
1511, 14nsyl 140 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
16 onsuc 7831 . . . . . . . 8 (𝐴 ∈ On → suc 𝐴 ∈ On)
177, 16syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → suc 𝐴 ∈ On)
18 sucidg 6465 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
197, 18syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
20 r1ord 9820 . . . . . . 7 (suc 𝐴 ∈ On → (𝐴 ∈ suc 𝐴 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴)))
2117, 19, 20sylc 65 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
22 r1elwf 9836 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
23 wfelirr 9865 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1 “ On) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
2421, 22, 233syl 18 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
25 simprr 773 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
2625fveq2d 6910 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = (𝑅1‘suc 𝑥))
27 r1suc 9810 . . . . . . . . 9 (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2827ad2antrl 728 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2926, 28eqtrd 2777 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = 𝒫 (𝑅1𝑥))
30 simplr 769 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ WUni)
317adantr 480 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 ∈ On)
32 sucidg 6465 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
3332ad2antrl 728 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
3433, 25eleqtrrd 2844 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥𝐴)
35 r1ord 9820 . . . . . . . . 9 (𝐴 ∈ On → (𝑥𝐴 → (𝑅1𝑥) ∈ (𝑅1𝐴)))
3631, 34, 35sylc 65 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝑥) ∈ (𝑅1𝐴))
3730, 36wunpw 10747 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1𝑥) ∈ (𝑅1𝐴))
3829, 37eqeltrd 2841 . . . . . 6 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ (𝑅1𝐴))
3938rexlimdvaa 3156 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1𝐴) ∈ (𝑅1𝐴)))
4024, 39mtod 198 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
41 ioran 986 . . . 4 (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
4215, 40, 41sylanbrc 583 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
43 dflim3 7868 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
449, 42, 43sylanbrc 583 . 2 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Lim 𝐴)
45 r1limwun 10776 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
4644, 45impbida 801 1 (𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))
Colors of variables: wff setvar class
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)
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