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Theorem r1limwun 9956
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 8999 . . 3 Tr (𝑅1𝐴)
21a1i 11 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → Tr (𝑅1𝐴))
3 limelon 6092 . . . . . 6 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4 r1fnon 8990 . . . . . . 7 𝑅1 Fn On
5 fndm 6288 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
64, 5ax-mp 5 . . . . . 6 dom 𝑅1 = On
73, 6syl6eleqr 2877 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
8 onssr1 9054 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
97, 8syl 17 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1𝐴))
10 0ellim 6091 . . . . 5 (Lim 𝐴 → ∅ ∈ 𝐴)
1110adantl 474 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
129, 11sseldd 3859 . . 3 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1𝐴))
1312ne0d 4187 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ≠ ∅)
14 rankuni 9086 . . . . . 6 (rank‘ 𝑥) = (rank‘𝑥)
15 rankon 9018 . . . . . . . . 9 (rank‘𝑥) ∈ On
16 eloni 6039 . . . . . . . . 9 ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
17 orduniss 6123 . . . . . . . . 9 (Ord (rank‘𝑥) → (rank‘𝑥) ⊆ (rank‘𝑥))
1815, 16, 17mp2b 10 . . . . . . . 8 (rank‘𝑥) ⊆ (rank‘𝑥)
1918a1i 11 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ⊆ (rank‘𝑥))
20 rankr1ai 9021 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → (rank‘𝑥) ∈ 𝐴)
2120adantl 474 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
22 onuni 7324 . . . . . . . . 9 ((rank‘𝑥) ∈ On → (rank‘𝑥) ∈ On)
2315, 22ax-mp 5 . . . . . . . 8 (rank‘𝑥) ∈ On
243adantr 473 . . . . . . . 8 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ On)
25 ontr2 6076 . . . . . . . 8 (( (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → (( (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → (rank‘𝑥) ∈ 𝐴))
2623, 24, 25sylancr 578 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (( (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → (rank‘𝑥) ∈ 𝐴))
2719, 21, 26mp2and 686 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
2814, 27syl5eqel 2870 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘ 𝑥) ∈ 𝐴)
29 r1elwf 9019 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → 𝑥 (𝑅1 “ On))
3029adantl 474 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
31 uniwf 9042 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ 𝑥 (𝑅1 “ On))
3230, 31sylib 210 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
337adantr 473 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
34 rankr1ag 9025 . . . . . 6 (( 𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3532, 33, 34syl2anc 576 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3628, 35mpbird 249 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 ∈ (𝑅1𝐴))
37 r1pwcl 9070 . . . . . 6 (Lim 𝐴 → (𝑥 ∈ (𝑅1𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1𝐴)))
3837adantl 474 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅1𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1𝐴)))
3938biimpa 469 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝒫 𝑥 ∈ (𝑅1𝐴))
4029ad2antlr 714 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
41 r1elwf 9019 . . . . . . . . 9 (𝑦 ∈ (𝑅1𝐴) → 𝑦 (𝑅1 “ On))
4241adantl 474 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑦 (𝑅1 “ On))
43 rankprb 9074 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
4440, 42, 43syl2anc 576 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
45 limord 6088 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
4645ad3antlr 718 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → Ord 𝐴)
4721adantr 473 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
48 rankr1ai 9021 . . . . . . . . . 10 (𝑦 ∈ (𝑅1𝐴) → (rank‘𝑦) ∈ 𝐴)
4948adantl 474 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑦) ∈ 𝐴)
50 ordunel 7358 . . . . . . . . 9 ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5146, 47, 49, 50syl3anc 1351 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
52 limsuc 7380 . . . . . . . . 9 (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5352ad3antlr 718 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5451, 53mpbid 224 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5544, 54eqeltrd 2866 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
56 prwf 9034 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5740, 42, 56syl2anc 576 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5833adantr 473 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
59 rankr1ag 9025 . . . . . . 7 (({𝑥, 𝑦} ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
6057, 58, 59syl2anc 576 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
6155, 60mpbird 249 . . . . 5 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → {𝑥, 𝑦} ∈ (𝑅1𝐴))
6261ralrimiva 3132 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴))
6336, 39, 623jca 1108 . . 3 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
6463ralrimiva 3132 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
65 fvex 6512 . . 3 (𝑅1𝐴) ∈ V
66 iswun 9924 . . 3 ((𝑅1𝐴) ∈ V → ((𝑅1𝐴) ∈ WUni ↔ (Tr (𝑅1𝐴) ∧ (𝑅1𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))))
6765, 66ax-mp 5 . 2 ((𝑅1𝐴) ∈ WUni ↔ (Tr (𝑅1𝐴) ∧ (𝑅1𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴))))
682, 13, 64, 67syl3anbrc 1323 1 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967  wral 3088  Vcvv 3415  cun 3827  wss 3829  c0 4178  𝒫 cpw 4422  {cpr 4443   cuni 4712  Tr wtr 5030  dom cdm 5407  cima 5410  Ord word 6028  Oncon0 6029  Lim wlim 6030  suc csuc 6031   Fn wfn 6183  cfv 6188  𝑅1cr1 8985  rankcrnk 8986  WUnicwun 9920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-reg 8851  ax-inf2 8898
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-r1 8987  df-rank 8988  df-wun 9922
This theorem is referenced by:  r1wunlim  9957  wunex3  9961
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