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Theorem r1limwun 9437
 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 8522 . . 3 Tr (𝑅1𝐴)
21a1i 11 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → Tr (𝑅1𝐴))
3 limelon 5705 . . . . . 6 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ On)
4 r1fnon 8513 . . . . . . 7 𝑅1 Fn On
5 fndm 5904 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
64, 5ax-mp 5 . . . . . 6 dom 𝑅1 = On
73, 6syl6eleqr 2699 . . . . 5 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
8 onssr1 8577 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
97, 8syl 17 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ⊆ (𝑅1𝐴))
10 0ellim 5704 . . . . 5 (Lim 𝐴 → ∅ ∈ 𝐴)
1110adantl 481 . . . 4 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ 𝐴)
129, 11sseldd 3569 . . 3 ((𝐴𝑉 ∧ Lim 𝐴) → ∅ ∈ (𝑅1𝐴))
13 ne0i 3880 . . 3 (∅ ∈ (𝑅1𝐴) → (𝑅1𝐴) ≠ ∅)
1412, 13syl 17 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ≠ ∅)
15 rankuni 8609 . . . . . 6 (rank‘ 𝑥) = (rank‘𝑥)
16 rankon 8541 . . . . . . . . 9 (rank‘𝑥) ∈ On
17 eloni 5650 . . . . . . . . 9 ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
18 orduniss 5738 . . . . . . . . 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 8544 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → (rank‘𝑥) ∈ 𝐴)
2221adantl 481 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
23 onuni 6885 . . . . . . . . 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 5689 . . . . . . . 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 711 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘𝑥) ∈ 𝐴)
2915, 28syl5eqel 2692 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → (rank‘ 𝑥) ∈ 𝐴)
30 r1elwf 8542 . . . . . . . 8 (𝑥 ∈ (𝑅1𝐴) → 𝑥 (𝑅1 “ On))
3130adantl 481 . . . . . . 7 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
32 uniwf 8565 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ 𝑥 (𝑅1 “ On))
3331, 32sylib 207 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
347adantr 480 . . . . . 6 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
35 rankr1ag 8548 . . . . . 6 (( 𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3633, 34, 35syl2anc 691 . . . . 5 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ↔ (rank‘ 𝑥) ∈ 𝐴))
3729, 36mpbird 246 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝑥 ∈ (𝑅1𝐴))
38 r1pwcl 8593 . . . . . 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 759 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
42 r1elwf 8542 . . . . . . . . 9 (𝑦 ∈ (𝑅1𝐴) → 𝑦 (𝑅1 “ On))
4342adantl 481 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → 𝑦 (𝑅1 “ On))
44 rankprb 8597 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
4541, 43, 44syl2anc 691 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) = suc ((rank‘𝑥) ∪ (rank‘𝑦)))
46 limord 5701 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
4746ad3antlr 763 . . . . . . . . 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 8544 . . . . . . . . . 10 (𝑦 ∈ (𝑅1𝐴) → (rank‘𝑦) ∈ 𝐴)
5049adantl 481 . . . . . . . . 9 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘𝑦) ∈ 𝐴)
51 ordunel 6919 . . . . . . . . 9 ((Ord 𝐴 ∧ (rank‘𝑥) ∈ 𝐴 ∧ (rank‘𝑦) ∈ 𝐴) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5247, 48, 50, 51syl3anc 1318 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
53 limsuc 6941 . . . . . . . . 9 (Lim 𝐴 → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5453ad3antlr 763 . . . . . . . 8 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴 ↔ suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴))
5552, 54mpbid 221 . . . . . . 7 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → suc ((rank‘𝑥) ∪ (rank‘𝑦)) ∈ 𝐴)
5645, 55eqeltrd 2688 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → (rank‘{𝑥, 𝑦}) ∈ 𝐴)
57 prwf 8557 . . . . . . . 8 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
5841, 43, 57syl2anc 691 . . . . . . 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 8548 . . . . . . 7 (({𝑥, 𝑦} ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
6158, 59, 60syl2anc 691 . . . . . 6 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → ({𝑥, 𝑦} ∈ (𝑅1𝐴) ↔ (rank‘{𝑥, 𝑦}) ∈ 𝐴))
6256, 61mpbird 246 . . . . 5 ((((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) ∧ 𝑦 ∈ (𝑅1𝐴)) → {𝑥, 𝑦} ∈ (𝑅1𝐴))
6362ralrimiva 2949 . . . 4 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴))
6437, 40, 633jca 1235 . . 3 (((𝐴𝑉 ∧ Lim 𝐴) ∧ 𝑥 ∈ (𝑅1𝐴)) → ( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
6564ralrimiva 2949 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → ∀𝑥 ∈ (𝑅1𝐴)( 𝑥 ∈ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴) ∧ ∀𝑦 ∈ (𝑅1𝐴){𝑥, 𝑦} ∈ (𝑅1𝐴)))
66 fvex 6113 . . 3 (𝑅1𝐴) ∈ V
67 iswun 9405 . . 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 1239 1 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {cpr 4127  ∪ cuni 4372  Tr wtr 4680  dom cdm 5038   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  𝑅1cr1 8508  rankcrnk 8509  WUnicwun 9401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511  df-wun 9403 This theorem is referenced by:  r1wunlim  9438  wunex3  9442
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