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Theorem r1limwun 9503
Description: Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
r1limwun ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)

Proof of Theorem r1limwun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1tr 8584 . . 3 Tr (𝑅1𝐴)
21a1i 11 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → Tr (𝑅1𝐴))
3 limelon 5750 . . . . . 6 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4 r1fnon 8575 . . . . . . 7 𝑅1 Fn On
5 fndm 5950 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
64, 5ax-mp 5 . . . . . 6 dom 𝑅1 = On
73, 6syl6eleqr 2715 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
8 onssr1 8639 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
97, 8syl 17 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1𝐴))
10 0ellim 5749 . . . . 5 (Lim 𝐴 → ∅ ∈ 𝐴)
1110adantl 482 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
129, 11sseldd 3589 . . 3 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1𝐴))
13 ne0i 3902 . . 3 (∅ ∈ (𝑅1𝐴) → (𝑅1𝐴) ≠ ∅)
1412, 13syl 17 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ≠ ∅)
15 rankuni 8671 . . . . . 6 (rank‘ 𝑥) = (rank‘𝑥)
16 rankon 8603 . . . . . . . . 9 (rank‘𝑥) ∈ On
17 eloni 5695 . . . . . . . . 9 ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
18 orduniss 5783 . . . . . . . . 9 (Ord (rank‘𝑥) → (rank‘𝑥) ⊆ (rank‘𝑥))
1916, 17, 18mp2b 10 . . . . . . . 8 (rank‘𝑥) ⊆ (rank‘𝑥)
2019a1i 11 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ⊆ (rank‘𝑥))
21 rankr1ai 8606 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → (rank‘𝑥) ∈ 𝐴)
2221adantl 482 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
23 onuni 6941 . . . . . . . . 9 ((rank‘𝑥) ∈ On → (rank‘𝑥) ∈ On)
2416, 23ax-mp 5 . . . . . . . 8 (rank‘𝑥) ∈ On
253adantr 481 . . . . . . . 8 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ On)
26 ontr2 5734 . . . . . . . 8 (( (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → (( (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → (rank‘𝑥) ∈ 𝐴))
2724, 25, 26sylancr 694 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (( (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → (rank‘𝑥) ∈ 𝐴))
2820, 22, 27mp2and 714 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
2915, 28syl5eqel 2708 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘ 𝑥) ∈ 𝐴)
30 r1elwf 8604 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → 𝑥 (𝑅1 “ On))
3130adantl 482 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
32 uniwf 8627 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ 𝑥 (𝑅1 “ On))
3331, 32sylib 208 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
347adantr 481 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
35 rankr1ag 8610 . . . . . 6 (( 𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3633, 34, 35syl2anc 692 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3729, 36mpbird 247 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 ∈ (𝑅1𝐴))
38 r1pwcl 8655 . . . . . 6 (Lim 𝐴 → (𝑥 ∈ (𝑅1𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1𝐴)))
3938adantl 482 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅1𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1𝐴)))
4039biimpa 501 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝒫 𝑥 ∈ (𝑅1𝐴))
4130ad2antlr 762 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
42 r1elwf 8604 . . . . . . . . 9 (𝑦 ∈ (𝑅1𝐴) → 𝑦 (𝑅1 “ On))
4342adantl 482 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑦 (𝑅1 “ On))
44 rankprb 8659 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
4541, 43, 44syl2anc 692 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
46 limord 5746 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
4746ad3antlr 766 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → Ord 𝐴)
4822adantr 481 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
49 rankr1ai 8606 . . . . . . . . . 10 (𝑦 ∈ (𝑅1𝐴) → (rank‘𝑦) ∈ 𝐴)
5049adantl 482 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑦) ∈ 𝐴)
51 ordunel 6975 . . . . . . . . 9 ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5247, 48, 50, 51syl3anc 1323 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
53 limsuc 6997 . . . . . . . . 9 (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5453ad3antlr 766 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5552, 54mpbid 222 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5645, 55eqeltrd 2704 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
57 prwf 8619 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5841, 43, 57syl2anc 692 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5934adantr 481 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
60 rankr1ag 8610 . . . . . . 7 (({𝑥, 𝑦} ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
6158, 59, 60syl2anc 692 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
6256, 61mpbird 247 . . . . 5 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → {𝑥, 𝑦} ∈ (𝑅1𝐴))
6362ralrimiva 2965 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴))
6437, 40, 633jca 1240 . . 3 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
6564ralrimiva 2965 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
66 fvex 6160 . . 3 (𝑅1𝐴) ∈ V
67 iswun 9471 . . 3 ((𝑅1𝐴) ∈ V → ((𝑅1𝐴) ∈ WUni ↔ (Tr (𝑅1𝐴) ∧ (𝑅1𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))))
6866, 67ax-mp 5 . 2 ((𝑅1𝐴) ∈ WUni ↔ (Tr (𝑅1𝐴) ∧ (𝑅1𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴))))
692, 14, 65, 68syl3anbrc 1244 1 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  Vcvv 3191  cun 3558  wss 3560  c0 3896  𝒫 cpw 4135  {cpr 4155   cuni 4407  Tr wtr 4717  dom cdm 5079  cima 5082  Ord word 5684  Oncon0 5685  Lim wlim 5686  suc csuc 5687   Fn wfn 5845  cfv 5850  𝑅1cr1 8570  rankcrnk 8571  WUnicwun 9467
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-reg 8442  ax-inf2 8483
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-r1 8572  df-rank 8573  df-wun 9469
This theorem is referenced by:  r1wunlim  9504  wunex3  9508
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