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Theorem r1wunlim 9438
 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 9419 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ∅ ∈ (𝑅1𝐴))
3 elfvdm 6130 . . . . . 6 (∅ ∈ (𝑅1𝐴) → 𝐴 ∈ dom 𝑅1)
42, 3syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5 r1fnon 8513 . . . . . 6 𝑅1 Fn On
6 fndm 5904 . . . . . 6 (𝑅1 Fn On → dom 𝑅1 = On)
75, 6ax-mp 5 . . . . 5 dom 𝑅1 = On
84, 7syl6eleq 2698 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ On)
9 eloni 5650 . . . 4 (𝐴 ∈ On → Ord 𝐴)
108, 9syl 17 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Ord 𝐴)
11 n0i 3879 . . . . . 6 (∅ ∈ (𝑅1𝐴) → ¬ (𝑅1𝐴) = ∅)
122, 11syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) = ∅)
13 fveq2 6103 . . . . . 6 (𝐴 = ∅ → (𝑅1𝐴) = (𝑅1‘∅))
14 r10 8514 . . . . . 6 (𝑅1‘∅) = ∅
1513, 14syl6eq 2660 . . . . 5 (𝐴 = ∅ → (𝑅1𝐴) = ∅)
1612, 15nsyl 134 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
17 suceloni 6905 . . . . . . . 8 (𝐴 ∈ On → suc 𝐴 ∈ On)
188, 17syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → suc 𝐴 ∈ On)
19 sucidg 5720 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
208, 19syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
21 r1ord 8526 . . . . . . 7 (suc 𝐴 ∈ On → (𝐴 ∈ suc 𝐴 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴)))
2218, 20, 21sylc 63 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
23 r1elwf 8542 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
24 wfelirr 8571 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1 “ On) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
2522, 23, 243syl 18 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
26 simprr 792 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
2726fveq2d 6107 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = (𝑅1‘suc 𝑥))
28 r1suc 8516 . . . . . . . . 9 (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2928ad2antrl 760 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
3027, 29eqtrd 2644 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = 𝒫 (𝑅1𝑥))
31 simplr 788 . . . . . . . 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 5720 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
3433ad2antrl 760 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
3534, 26eleqtrrd 2691 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥𝐴)
36 r1ord 8526 . . . . . . . . 9 (𝐴 ∈ On → (𝑥𝐴 → (𝑅1𝑥) ∈ (𝑅1𝐴)))
3732, 35, 36sylc 63 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝑥) ∈ (𝑅1𝐴))
3831, 37wunpw 9408 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1𝑥) ∈ (𝑅1𝐴))
3930, 38eqeltrd 2688 . . . . . 6 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ (𝑅1𝐴))
4039rexlimdvaa 3014 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1𝐴) ∈ (𝑅1𝐴)))
4125, 40mtod 188 . . . 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 695 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
44 dflim3 6939 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
4510, 43, 44sylanbrc 695 . 2 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Lim 𝐴)
46 r1limwun 9437 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
4745, 46impbida 873 1 (𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  𝑅1cr1 8508  WUnicwun 9401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511  df-wun 9403 This theorem is referenced by: (None)
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