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Theorem r1limwun 10761
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 9801 . . 3 Tr (𝑅1𝐴)
21a1i 11 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → Tr (𝑅1𝐴))
3 limelon 6435 . . . . . 6 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4 r1fnon 9792 . . . . . . 7 𝑅1 Fn On
54fndmi 6659 . . . . . 6 dom 𝑅1 = On
63, 5eleqtrrdi 2836 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
7 onssr1 9856 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
86, 7syl 17 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1𝐴))
9 0ellim 6434 . . . . 5 (Lim 𝐴 → ∅ ∈ 𝐴)
109adantl 480 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
118, 10sseldd 3977 . . 3 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1𝐴))
1211ne0d 4335 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ≠ ∅)
13 rankuni 9888 . . . . . 6 (rank‘ 𝑥) = (rank‘𝑥)
14 rankon 9820 . . . . . . . . 9 (rank‘𝑥) ∈ On
15 eloni 6381 . . . . . . . . 9 ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
16 orduniss 6468 . . . . . . . . 9 (Ord (rank‘𝑥) → (rank‘𝑥) ⊆ (rank‘𝑥))
1714, 15, 16mp2b 10 . . . . . . . 8 (rank‘𝑥) ⊆ (rank‘𝑥)
1817a1i 11 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ⊆ (rank‘𝑥))
19 rankr1ai 9823 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → (rank‘𝑥) ∈ 𝐴)
2019adantl 480 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
21 onuni 7792 . . . . . . . . 9 ((rank‘𝑥) ∈ On → (rank‘𝑥) ∈ On)
2214, 21ax-mp 5 . . . . . . . 8 (rank‘𝑥) ∈ On
233adantr 479 . . . . . . . 8 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ On)
24 ontr2 6418 . . . . . . . 8 (( (rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → (( (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → (rank‘𝑥) ∈ 𝐴))
2522, 23, 24sylancr 585 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (( (rank‘𝑥) ⊆ (rank‘𝑥) ∧ (rank‘𝑥) ∈ 𝐴) → (rank‘𝑥) ∈ 𝐴))
2618, 20, 25mp2and 697 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
2713, 26eqeltrid 2829 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘ 𝑥) ∈ 𝐴)
28 r1elwf 9821 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → 𝑥 (𝑅1 “ On))
2928adantl 480 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
30 uniwf 9844 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ 𝑥 (𝑅1 “ On))
3129, 30sylib 217 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
326adantr 479 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
33 rankr1ag 9827 . . . . . 6 (( 𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3431, 32, 33syl2anc 582 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3527, 34mpbird 256 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 ∈ (𝑅1𝐴))
36 r1pwcl 9872 . . . . . 6 (Lim 𝐴 → (𝑥 ∈ (𝑅1𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1𝐴)))
3736adantl 480 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑥 ∈ (𝑅1𝐴) ↔ 𝒫 𝑥 ∈ (𝑅1𝐴)))
3837biimpa 475 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝒫 𝑥 ∈ (𝑅1𝐴))
3928ad2antlr 725 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
40 r1elwf 9821 . . . . . . . . 9 (𝑦 ∈ (𝑅1𝐴) → 𝑦 (𝑅1 “ On))
4140adantl 480 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑦 (𝑅1 “ On))
42 rankprb 9876 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
4339, 41, 42syl2anc 582 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
44 limord 6431 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
4544ad3antlr 729 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → Ord 𝐴)
4620adantr 479 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
47 rankr1ai 9823 . . . . . . . . . 10 (𝑦 ∈ (𝑅1𝐴) → (rank‘𝑦) ∈ 𝐴)
4847adantl 480 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑦) ∈ 𝐴)
49 ordunel 7831 . . . . . . . . 9 ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5045, 46, 48, 49syl3anc 1368 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
51 limsuc 7854 . . . . . . . . 9 (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5251ad3antlr 729 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5350, 52mpbid 231 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5443, 53eqeltrd 2825 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
55 prwf 9836 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5639, 41, 55syl2anc 582 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5732adantr 479 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
58 rankr1ag 9827 . . . . . . 7 (({𝑥, 𝑦} ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
5956, 57, 58syl2anc 582 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
6054, 59mpbird 256 . . . . 5 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → {𝑥, 𝑦} ∈ (𝑅1𝐴))
6160ralrimiva 3135 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴))
6235, 38, 613jca 1125 . . 3 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
6362ralrimiva 3135 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
64 fvex 6909 . . 3 (𝑅1𝐴) ∈ V
65 iswun 10729 . . 3 ((𝑅1𝐴) ∈ V → ((𝑅1𝐴) ∈ WUni ↔ (Tr (𝑅1𝐴) ∧ (𝑅1𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))))
6664, 65ax-mp 5 . 2 ((𝑅1𝐴) ∈ WUni ↔ (Tr (𝑅1𝐴) ∧ (𝑅1𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴))))
672, 12, 63, 66syl3anbrc 1340 1 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  Vcvv 3461  cun 3942  wss 3944  c0 4322  𝒫 cpw 4604  {cpr 4632   cuni 4909  Tr wtr 5266  dom cdm 5678  cima 5681  Ord word 6370  Oncon0 6371  Lim wlim 6372  suc csuc 6373  cfv 6549  𝑅1cr1 9787  rankcrnk 9788  WUnicwun 10725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-reg 9617  ax-inf2 9666
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-r1 9789  df-rank 9790  df-wun 10727
This theorem is referenced by:  r1wunlim  10762  wunex3  10766
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