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Theorem inatsk 9544
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 9456 . . . . . 6 (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2 winaon 9454 . . . . . . . . . 10 (𝐴 ∈ Inaccw𝐴 ∈ On)
3 winalim 9461 . . . . . . . . . 10 (𝐴 ∈ Inaccw → Lim 𝐴)
4 r1lim 8579 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
52, 3, 4syl2anc 692 . . . . . . . . 9 (𝐴 ∈ Inaccw → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
65eleq2d 2684 . . . . . . . 8 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ 𝑥 𝑦𝐴 (𝑅1𝑦)))
7 eliun 4490 . . . . . . . 8 (𝑥 𝑦𝐴 (𝑅1𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦))
86, 7syl6bb 276 . . . . . . 7 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦)))
9 onelon 5707 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
102, 9sylan 488 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → 𝑦 ∈ On)
11 r1pw 8652 . . . . . . . . . 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 6996 . . . . . . . . . . . . 13 (Lim 𝐴 → (𝑦𝐴 ↔ suc 𝑦𝐴))
143, 13syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (𝑦𝐴 ↔ suc 𝑦𝐴))
15 r1ord2 8588 . . . . . . . . . . . . 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 445 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴))
1918sseld 3582 . . . . . . . . 9 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2012, 19sylbid 230 . . . . . . . 8 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2120rexlimdva 3024 . . . . . . 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 445 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝒫 𝑥 ∈ (𝑅1𝐴))
25 elssuni 4433 . . . . 5 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 (𝑅1𝐴))
26 r1tr2 8584 . . . . 5 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2725, 26syl6ss 3595 . . . 4 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ⊆ (𝑅1𝐴))
2824, 27jccil 562 . . 3 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → (𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
2928ralrimiva 2960 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
301, 2syl 17 . . . . . . . . 9 (𝐴 ∈ Inacc → 𝐴 ∈ On)
31 r1suc 8577 . . . . . . . . . 10 (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
3231eleq2d 2684 . . . . . . . . 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 8605 . . . . . . . 8 (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
3533, 34syl6bir 244 . . . . . . 7 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → (rank‘𝑥) ∈ suc 𝐴))
3635imp 445 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37 fvex 6158 . . . . . . 7 (rank‘𝑥) ∈ V
3837elsuc 5753 . . . . . 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 403 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41 fvex 6158 . . . . . . . 8 (𝑅1𝐴) ∈ V
42 elpwi 4140 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 ⊆ (𝑅1𝐴))
4342ad2antlr 762 . . . . . . . 8 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1𝐴))
44 ssdomg 7945 . . . . . . . 8 ((𝑅1𝐴) ∈ V → (𝑥 ⊆ (𝑅1𝐴) → 𝑥 ≼ (𝑅1𝐴)))
4541, 43, 44mpsyl 68 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≼ (𝑅1𝐴))
46 rankcf 9543 . . . . . . . . . 10 ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47 fveq2 6148 . . . . . . . . . . . 12 ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48 elina 9453 . . . . . . . . . . . . 13 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
4948simp2bi 1075 . . . . . . . . . . . 12 (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
5047, 49sylan9eqr 2677 . . . . . . . . . . 11 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
5150breq2d 4625 . . . . . . . . . 10 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥𝐴))
5246, 51mtbii 316 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥𝐴)
53 inar1 9541 . . . . . . . . . . 11 (𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)
54 sdomentr 8038 . . . . . . . . . . . 12 ((𝑥 ≺ (𝑅1𝐴) ∧ (𝑅1𝐴) ≈ 𝐴) → 𝑥𝐴)
5554expcom 451 . . . . . . . . . . 11 ((𝑅1𝐴) ≈ 𝐴 → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5653, 55syl 17 . . . . . . . . . 10 (𝐴 ∈ Inacc → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5756adantr 481 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5852, 57mtod 189 . . . . . . . 8 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
5958adantlr 750 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
60 bren2 7930 . . . . . . 7 (𝑥 ≈ (𝑅1𝐴) ↔ (𝑥 ≼ (𝑅1𝐴) ∧ ¬ 𝑥 ≺ (𝑅1𝐴)))
6145, 59, 60sylanbrc 697 . . . . . 6 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅1𝐴))
6261ex 450 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) = 𝐴𝑥 ≈ (𝑅1𝐴)))
63 r1elwf 8603 . . . . . . . . 9 (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 (𝑅1 “ On))
6433, 63syl6bir 244 . . . . . . . 8 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 (𝑅1 “ On)))
6564imp 445 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
66 r1fnon 8574 . . . . . . . . . 10 𝑅1 Fn On
67 fndm 5948 . . . . . . . . . 10 (𝑅1 Fn On → dom 𝑅1 = On)
6866, 67ax-mp 5 . . . . . . . . 9 dom 𝑅1 = On
6930, 68syl6eleqr 2709 . . . . . . . 8 (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
7069adantr 481 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
71 rankr1ag 8609 . . . . . . 7 ((𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7265, 70, 71syl2anc 692 . . . . . 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 882 . . . 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 2960 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))
77 eltsk2g 9517 . . 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 697 1 (𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   ciun 4485   class class class wbr 4613  dom cdm 5074  cima 5077  Oncon0 5682  Lim wlim 5683  suc csuc 5684   Fn wfn 5842  cfv 5847  cen 7896  cdom 7897  csdm 7898  𝑅1cr1 8569  rankcrnk 8570  cfccf 8707  Inaccwcwina 9448  Inacccina 9449  Tarskictsk 9514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-ac2 9229
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-oi 8359  df-r1 8571  df-rank 8572  df-card 8709  df-cf 8711  df-acn 8712  df-ac 8883  df-wina 9450  df-ina 9451  df-tsk 9515
This theorem is referenced by:  r1omtsk  9545  r1tskina  9548  grutsk  9588
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