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Theorem r1wunlim 9503
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 477 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (𝑅1𝐴) ∈ WUni)
21wun0 9484 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ∅ ∈ (𝑅1𝐴))
3 elfvdm 6177 . . . . . 6 (∅ ∈ (𝑅1𝐴) → 𝐴 ∈ dom 𝑅1)
42, 3syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5 r1fnon 8574 . . . . . 6 𝑅1 Fn On
6 fndm 5948 . . . . . 6 (𝑅1 Fn On → dom 𝑅1 = On)
75, 6ax-mp 5 . . . . 5 dom 𝑅1 = On
84, 7syl6eleq 2708 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ On)
9 eloni 5692 . . . 4 (𝐴 ∈ On → Ord 𝐴)
108, 9syl 17 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Ord 𝐴)
11 n0i 3896 . . . . . 6 (∅ ∈ (𝑅1𝐴) → ¬ (𝑅1𝐴) = ∅)
122, 11syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) = ∅)
13 fveq2 6148 . . . . . 6 (𝐴 = ∅ → (𝑅1𝐴) = (𝑅1‘∅))
14 r10 8575 . . . . . 6 (𝑅1‘∅) = ∅
1513, 14syl6eq 2671 . . . . 5 (𝐴 = ∅ → (𝑅1𝐴) = ∅)
1612, 15nsyl 135 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
17 suceloni 6960 . . . . . . . 8 (𝐴 ∈ On → suc 𝐴 ∈ On)
188, 17syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → suc 𝐴 ∈ On)
19 sucidg 5762 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
208, 19syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
21 r1ord 8587 . . . . . . 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 8603 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
24 wfelirr 8632 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1 “ On) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
2522, 23, 243syl 18 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
26 simprr 795 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
2726fveq2d 6152 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = (𝑅1‘suc 𝑥))
28 r1suc 8577 . . . . . . . . 9 (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2928ad2antrl 763 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
3027, 29eqtrd 2655 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = 𝒫 (𝑅1𝑥))
31 simplr 791 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ WUni)
328adantr 481 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 ∈ On)
33 sucidg 5762 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
3433ad2antrl 763 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
3534, 26eleqtrrd 2701 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥𝐴)
36 r1ord 8587 . . . . . . . . 9 (𝐴 ∈ On → (𝑥𝐴 → (𝑅1𝑥) ∈ (𝑅1𝐴)))
3732, 35, 36sylc 65 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝑥) ∈ (𝑅1𝐴))
3831, 37wunpw 9473 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1𝑥) ∈ (𝑅1𝐴))
3930, 38eqeltrd 2698 . . . . . 6 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ (𝑅1𝐴))
4039rexlimdvaa 3025 . . . . 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 511 . . . 4 (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
4316, 41, 42sylanbrc 697 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
44 dflim3 6994 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
4510, 43, 44sylanbrc 697 . 2 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Lim 𝐴)
46 r1limwun 9502 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
4745, 46impbida 876 1 (𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wrex 2908  c0 3891  𝒫 cpw 4130   cuni 4402  dom cdm 5074  cima 5077  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684   Fn wfn 5842  cfv 5847  𝑅1cr1 8569  WUnicwun 9466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-r1 8571  df-rank 8572  df-wun 9468
This theorem is referenced by: (None)
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