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Theorem inatsk 9638
Description: (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.)
Assertion
Ref Expression
inatsk (𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)

Proof of Theorem inatsk
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inawina 9550 . . . . . 6 (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2 winaon 9548 . . . . . . . . . 10 (𝐴 ∈ Inaccw𝐴 ∈ On)
3 winalim 9555 . . . . . . . . . 10 (𝐴 ∈ Inaccw → Lim 𝐴)
4 r1lim 8673 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
52, 3, 4syl2anc 694 . . . . . . . . 9 (𝐴 ∈ Inaccw → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
65eleq2d 2716 . . . . . . . 8 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ 𝑥 𝑦𝐴 (𝑅1𝑦)))
7 eliun 4556 . . . . . . . 8 (𝑥 𝑦𝐴 (𝑅1𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦))
86, 7syl6bb 276 . . . . . . 7 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦)))
9 onelon 5786 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
102, 9sylan 487 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → 𝑦 ∈ On)
11 r1pw 8746 . . . . . . . . . 10 (𝑦 ∈ On → (𝑥 ∈ (𝑅1𝑦) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝑦)))
1210, 11syl 17 . . . . . . . . 9 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑥 ∈ (𝑅1𝑦) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝑦)))
13 limsuc 7091 . . . . . . . . . . . . 13 (Lim 𝐴 → (𝑦𝐴 ↔ suc 𝑦𝐴))
143, 13syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (𝑦𝐴 ↔ suc 𝑦𝐴))
15 r1ord2 8682 . . . . . . . . . . . . 13 (𝐴 ∈ On → (suc 𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
162, 15syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (suc 𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
1714, 16sylbid 230 . . . . . . . . . . 11 (𝐴 ∈ Inaccw → (𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
1817imp 444 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴))
1918sseld 3635 . . . . . . . . 9 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2012, 19sylbid 230 . . . . . . . 8 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2120rexlimdva 3060 . . . . . . 7 (𝐴 ∈ Inaccw → (∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
228, 21sylbid 230 . . . . . 6 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
231, 22syl 17 . . . . 5 (𝐴 ∈ Inacc → (𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2423imp 444 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝒫 𝑥 ∈ (𝑅1𝐴))
25 elssuni 4499 . . . . 5 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 (𝑅1𝐴))
26 r1tr2 8678 . . . . 5 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2725, 26syl6ss 3648 . . . 4 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ⊆ (𝑅1𝐴))
2824, 27jccil 562 . . 3 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → (𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
2928ralrimiva 2995 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
301, 2syl 17 . . . . . . . . 9 (𝐴 ∈ Inacc → 𝐴 ∈ On)
31 r1suc 8671 . . . . . . . . . 10 (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
3231eleq2d 2716 . . . . . . . . 9 (𝐴 ∈ On → (𝑥 ∈ (𝑅1‘suc 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑅1𝐴)))
3330, 32syl 17 . . . . . . . 8 (𝐴 ∈ Inacc → (𝑥 ∈ (𝑅1‘suc 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑅1𝐴)))
34 rankr1ai 8699 . . . . . . . 8 (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
3533, 34syl6bir 244 . . . . . . 7 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → (rank‘𝑥) ∈ suc 𝐴))
3635imp 444 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37 fvex 6239 . . . . . . 7 (rank‘𝑥) ∈ V
3837elsuc 5832 . . . . . 6 ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
3936, 38sylib 208 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
4039orcomd 402 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41 fvex 6239 . . . . . . . 8 (𝑅1𝐴) ∈ V
42 elpwi 4201 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 ⊆ (𝑅1𝐴))
4342ad2antlr 763 . . . . . . . 8 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1𝐴))
44 ssdomg 8043 . . . . . . . 8 ((𝑅1𝐴) ∈ V → (𝑥 ⊆ (𝑅1𝐴) → 𝑥 ≼ (𝑅1𝐴)))
4541, 43, 44mpsyl 68 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≼ (𝑅1𝐴))
46 rankcf 9637 . . . . . . . . . 10 ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47 fveq2 6229 . . . . . . . . . . . 12 ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48 elina 9547 . . . . . . . . . . . . 13 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
4948simp2bi 1097 . . . . . . . . . . . 12 (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
5047, 49sylan9eqr 2707 . . . . . . . . . . 11 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
5150breq2d 4697 . . . . . . . . . 10 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥𝐴))
5246, 51mtbii 315 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥𝐴)
53 inar1 9635 . . . . . . . . . . 11 (𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)
54 sdomentr 8135 . . . . . . . . . . . 12 ((𝑥 ≺ (𝑅1𝐴) ∧ (𝑅1𝐴) ≈ 𝐴) → 𝑥𝐴)
5554expcom 450 . . . . . . . . . . 11 ((𝑅1𝐴) ≈ 𝐴 → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5653, 55syl 17 . . . . . . . . . 10 (𝐴 ∈ Inacc → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5756adantr 480 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5852, 57mtod 189 . . . . . . . 8 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
5958adantlr 751 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
60 bren2 8028 . . . . . . 7 (𝑥 ≈ (𝑅1𝐴) ↔ (𝑥 ≼ (𝑅1𝐴) ∧ ¬ 𝑥 ≺ (𝑅1𝐴)))
6145, 59, 60sylanbrc 699 . . . . . 6 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅1𝐴))
6261ex 449 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) = 𝐴𝑥 ≈ (𝑅1𝐴)))
63 r1elwf 8697 . . . . . . . . 9 (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 (𝑅1 “ On))
6433, 63syl6bir 244 . . . . . . . 8 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 (𝑅1 “ On)))
6564imp 444 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
66 r1fnon 8668 . . . . . . . . . 10 𝑅1 Fn On
67 fndm 6028 . . . . . . . . . 10 (𝑅1 Fn On → dom 𝑅1 = On)
6866, 67ax-mp 5 . . . . . . . . 9 dom 𝑅1 = On
6930, 68syl6eleqr 2741 . . . . . . . 8 (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
7069adantr 480 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
71 rankr1ag 8703 . . . . . . 7 ((𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7265, 70, 71syl2anc 694 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7372biimprd 238 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) ∈ 𝐴𝑥 ∈ (𝑅1𝐴)))
7462, 73orim12d 901 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴) → (𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴))))
7540, 74mpd 15 . . 3 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))
7675ralrimiva 2995 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))
77 eltsk2g 9611 . . 3 ((𝑅1𝐴) ∈ V → ((𝑅1𝐴) ∈ Tarski ↔ (∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)) ∧ ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))))
7841, 77ax-mp 5 . 2 ((𝑅1𝐴) ∈ Tarski ↔ (∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)) ∧ ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴))))
7929, 76, 78sylanbrc 699 1 (𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   ciun 4552   class class class wbr 4685  dom cdm 5143  cima 5146  Oncon0 5761  Lim wlim 5762  suc csuc 5763   Fn wfn 5921  cfv 5926  cen 7994  cdom 7995  csdm 7996  𝑅1cr1 8663  rankcrnk 8664  cfccf 8801  Inaccwcwina 9542  Inacccina 9543  Tarskictsk 9608
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-inf2 8576  ax-ac2 9323
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-rmo 2949  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-iin 4555  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-se 5103  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-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-r1 8665  df-rank 8666  df-card 8803  df-cf 8805  df-acn 8806  df-ac 8977  df-wina 9544  df-ina 9545  df-tsk 9609
This theorem is referenced by:  r1omtsk  9639  r1tskina  9642  grutsk  9682
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