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Theorem r1limwun 9596
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 8677 . . 3 Tr (𝑅1𝐴)
21a1i 11 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → Tr (𝑅1𝐴))
3 limelon 5826 . . . . . 6 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4 r1fnon 8668 . . . . . . 7 𝑅1 Fn On
5 fndm 6028 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
64, 5ax-mp 5 . . . . . 6 dom 𝑅1 = On
73, 6syl6eleqr 2741 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
8 onssr1 8732 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
97, 8syl 17 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1𝐴))
10 0ellim 5825 . . . . 5 (Lim 𝐴 → ∅ ∈ 𝐴)
1110adantl 481 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
129, 11sseldd 3637 . . 3 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1𝐴))
13 ne0i 3954 . . 3 (∅ ∈ (𝑅1𝐴) → (𝑅1𝐴) ≠ ∅)
1412, 13syl 17 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ≠ ∅)
15 rankuni 8764 . . . . . 6 (rank‘ 𝑥) = (rank‘𝑥)
16 rankon 8696 . . . . . . . . 9 (rank‘𝑥) ∈ On
17 eloni 5771 . . . . . . . . 9 ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
18 orduniss 5859 . . . . . . . . 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 8699 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → (rank‘𝑥) ∈ 𝐴)
2221adantl 481 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
23 onuni 7035 . . . . . . . . 9 ((rank‘𝑥) ∈ On → (rank‘𝑥) ∈ On)
2416, 23ax-mp 5 . . . . . . . 8 (rank‘𝑥) ∈ On
253adantr 480 . . . . . . . 8 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ On)
26 ontr2 5810 . . . . . . . 8 (( (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → (( (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → (rank‘𝑥) ∈ 𝐴))
2724, 25, 26sylancr 696 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (( (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → (rank‘𝑥) ∈ 𝐴))
2820, 22, 27mp2and 715 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
2915, 28syl5eqel 2734 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘ 𝑥) ∈ 𝐴)
30 r1elwf 8697 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → 𝑥 (𝑅1 “ On))
3130adantl 481 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
32 uniwf 8720 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ 𝑥 (𝑅1 “ On))
3331, 32sylib 208 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
347adantr 480 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
35 rankr1ag 8703 . . . . . 6 (( 𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3633, 34, 35syl2anc 694 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3729, 36mpbird 247 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 ∈ (𝑅1𝐴))
38 r1pwcl 8748 . . . . . 6 (Lim 𝐴 → (𝑥 ∈ (𝑅1𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1𝐴)))
3938adantl 481 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅1𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1𝐴)))
4039biimpa 500 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝒫 𝑥 ∈ (𝑅1𝐴))
4130ad2antlr 763 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
42 r1elwf 8697 . . . . . . . . 9 (𝑦 ∈ (𝑅1𝐴) → 𝑦 (𝑅1 “ On))
4342adantl 481 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑦 (𝑅1 “ On))
44 rankprb 8752 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
4541, 43, 44syl2anc 694 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
46 limord 5822 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
4746ad3antlr 767 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → Ord 𝐴)
4822adantr 480 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
49 rankr1ai 8699 . . . . . . . . . 10 (𝑦 ∈ (𝑅1𝐴) → (rank‘𝑦) ∈ 𝐴)
5049adantl 481 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑦) ∈ 𝐴)
51 ordunel 7069 . . . . . . . . 9 ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5247, 48, 50, 51syl3anc 1366 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
53 limsuc 7091 . . . . . . . . 9 (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5453ad3antlr 767 . . . . . . . 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 2730 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
57 prwf 8712 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5841, 43, 57syl2anc 694 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5934adantr 480 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
60 rankr1ag 8703 . . . . . . 7 (({𝑥, 𝑦} ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
6158, 59, 60syl2anc 694 . . . . . 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 2995 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴))
6437, 40, 633jca 1261 . . 3 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
6564ralrimiva 2995 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
66 fvex 6239 . . 3 (𝑅1𝐴) ∈ V
67 iswun 9564 . . 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 1265 1 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cun 3605  wss 3607  c0 3948  𝒫 cpw 4191  {cpr 4212   cuni 4468  Tr wtr 4785  dom cdm 5143  cima 5146  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763   Fn wfn 5921  cfv 5926  𝑅1cr1 8663  rankcrnk 8664  WUnicwun 9560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666  df-wun 9562
This theorem is referenced by:  r1wunlim  9597  wunex3  9601
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