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Theorem inatsk 9479
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 9391 . . . . . 6 (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2 winaon 9389 . . . . . . . . . 10 (𝐴 ∈ Inaccw𝐴 ∈ On)
3 winalim 9396 . . . . . . . . . 10 (𝐴 ∈ Inaccw → Lim 𝐴)
4 r1lim 8518 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
52, 3, 4syl2anc 691 . . . . . . . . 9 (𝐴 ∈ Inaccw → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
65eleq2d 2673 . . . . . . . 8 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ 𝑥 𝑦𝐴 (𝑅1𝑦)))
7 eliun 4460 . . . . . . . 8 (𝑥 𝑦𝐴 (𝑅1𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦))
86, 7syl6bb 275 . . . . . . 7 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦)))
9 onelon 5665 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
102, 9sylan 487 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → 𝑦 ∈ On)
11 r1pw 8591 . . . . . . . . . 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 6941 . . . . . . . . . . . . 13 (Lim 𝐴 → (𝑦𝐴 ↔ suc 𝑦𝐴))
143, 13syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (𝑦𝐴 ↔ suc 𝑦𝐴))
15 r1ord2 8527 . . . . . . . . . . . . 13 (𝐴 ∈ On → (suc 𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
162, 15syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (suc 𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
1714, 16sylbid 229 . . . . . . . . . . 11 (𝐴 ∈ Inaccw → (𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
1817imp 444 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴))
1918sseld 3567 . . . . . . . . 9 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2012, 19sylbid 229 . . . . . . . 8 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2120rexlimdva 3013 . . . . . . 7 (𝐴 ∈ Inaccw → (∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
228, 21sylbid 229 . . . . . 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 4403 . . . . 5 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 (𝑅1𝐴))
26 r1tr2 8523 . . . . 5 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2725, 26syl6ss 3580 . . . 4 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ⊆ (𝑅1𝐴))
2824, 27jccil 561 . . 3 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → (𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
2928ralrimiva 2949 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
301, 2syl 17 . . . . . . . . 9 (𝐴 ∈ Inacc → 𝐴 ∈ On)
31 r1suc 8516 . . . . . . . . . 10 (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
3231eleq2d 2673 . . . . . . . . 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 8544 . . . . . . . 8 (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
3533, 34syl6bir 243 . . . . . . 7 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → (rank‘𝑥) ∈ suc 𝐴))
3635imp 444 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37 fvex 6113 . . . . . . 7 (rank‘𝑥) ∈ V
3837elsuc 5711 . . . . . 6 ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
3936, 38sylib 207 . . . . 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 6113 . . . . . . . 8 (𝑅1𝐴) ∈ V
42 elpwi 4117 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 ⊆ (𝑅1𝐴))
4342ad2antlr 759 . . . . . . . 8 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1𝐴))
44 ssdomg 7887 . . . . . . . 8 ((𝑅1𝐴) ∈ V → (𝑥 ⊆ (𝑅1𝐴) → 𝑥 ≼ (𝑅1𝐴)))
4541, 43, 44mpsyl 66 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≼ (𝑅1𝐴))
46 rankcf 9478 . . . . . . . . . 10 ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47 fveq2 6103 . . . . . . . . . . . 12 ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48 elina 9388 . . . . . . . . . . . . 13 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
4948simp2bi 1070 . . . . . . . . . . . 12 (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
5047, 49sylan9eqr 2666 . . . . . . . . . . 11 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
5150breq2d 4595 . . . . . . . . . 10 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥𝐴))
5246, 51mtbii 315 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥𝐴)
53 inar1 9476 . . . . . . . . . . 11 (𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)
54 sdomentr 7979 . . . . . . . . . . . 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 188 . . . . . . . 8 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
5958adantlr 747 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
60 bren2 7872 . . . . . . 7 (𝑥 ≈ (𝑅1𝐴) ↔ (𝑥 ≼ (𝑅1𝐴) ∧ ¬ 𝑥 ≺ (𝑅1𝐴)))
6145, 59, 60sylanbrc 695 . . . . . 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 8542 . . . . . . . . 9 (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 (𝑅1 “ On))
6433, 63syl6bir 243 . . . . . . . 8 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 (𝑅1 “ On)))
6564imp 444 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
66 r1fnon 8513 . . . . . . . . . 10 𝑅1 Fn On
67 fndm 5904 . . . . . . . . . 10 (𝑅1 Fn On → dom 𝑅1 = On)
6866, 67ax-mp 5 . . . . . . . . 9 dom 𝑅1 = On
6930, 68syl6eleqr 2699 . . . . . . . 8 (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
7069adantr 480 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
71 rankr1ag 8548 . . . . . . 7 ((𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7265, 70, 71syl2anc 691 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7372biimprd 237 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) ∈ 𝐴𝑥 ∈ (𝑅1𝐴)))
7462, 73orim12d 879 . . . 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 2949 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))
77 eltsk2g 9452 . . 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 695 1 (𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  Vcvv 3173  wss 3540  c0 3874  𝒫 cpw 4108   cuni 4372   ciun 4455   class class class wbr 4583  dom cdm 5038  cima 5041  Oncon0 5640  Lim wlim 5641  suc csuc 5642   Fn wfn 5799  cfv 5804  cen 7838  cdom 7839  csdm 7840  𝑅1cr1 8508  rankcrnk 8509  cfccf 8646  Inaccwcwina 9383  Inacccina 9384  Tarskictsk 9449
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-inf2 8421  ax-ac2 9168
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-rmo 2904  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-iin 4458  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-se 4998  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-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-r1 8510  df-rank 8511  df-card 8648  df-cf 8650  df-acn 8651  df-ac 8822  df-wina 9385  df-ina 9386  df-tsk 9450
This theorem is referenced by:  r1omtsk  9480  r1tskina  9483  grutsk  9523
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