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Theorem inatsk 10810
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 10722 . . . . . 6 (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2 winaon 10720 . . . . . . . . . 10 (𝐴 ∈ Inaccw𝐴 ∈ On)
3 winalim 10727 . . . . . . . . . 10 (𝐴 ∈ Inaccw → Lim 𝐴)
4 r1lim 9806 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
52, 3, 4syl2anc 582 . . . . . . . . 9 (𝐴 ∈ Inaccw → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
65eleq2d 2812 . . . . . . . 8 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ 𝑥 𝑦𝐴 (𝑅1𝑦)))
7 eliun 4998 . . . . . . . 8 (𝑥 𝑦𝐴 (𝑅1𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦))
86, 7bitrdi 286 . . . . . . 7 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦)))
9 onelon 6391 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
102, 9sylan 578 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → 𝑦 ∈ On)
11 r1pw 9879 . . . . . . . . . 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 7849 . . . . . . . . . . . . 13 (Lim 𝐴 → (𝑦𝐴 ↔ suc 𝑦𝐴))
143, 13syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (𝑦𝐴 ↔ suc 𝑦𝐴))
15 r1ord2 9815 . . . . . . . . . . . . 13 (𝐴 ∈ On → (suc 𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
162, 15syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (suc 𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
1714, 16sylbid 239 . . . . . . . . . . 11 (𝐴 ∈ Inaccw → (𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
1817imp 405 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴))
1918sseld 3978 . . . . . . . . 9 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2012, 19sylbid 239 . . . . . . . 8 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2120rexlimdva 3145 . . . . . . 7 (𝐴 ∈ Inaccw → (∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
228, 21sylbid 239 . . . . . 6 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
231, 22syl 17 . . . . 5 (𝐴 ∈ Inacc → (𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2423imp 405 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝒫 𝑥 ∈ (𝑅1𝐴))
25 elssuni 4938 . . . . 5 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 (𝑅1𝐴))
26 r1tr2 9811 . . . . 5 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2725, 26sstrdi 3992 . . . 4 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ⊆ (𝑅1𝐴))
2824, 27jccil 521 . . 3 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → (𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
2928ralrimiva 3136 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
301, 2syl 17 . . . . . . . . 9 (𝐴 ∈ Inacc → 𝐴 ∈ On)
31 r1suc 9804 . . . . . . . . . 10 (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
3231eleq2d 2812 . . . . . . . . 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 9832 . . . . . . . 8 (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
3533, 34biimtrrdi 253 . . . . . . 7 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → (rank‘𝑥) ∈ suc 𝐴))
3635imp 405 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37 fvex 6904 . . . . . . 7 (rank‘𝑥) ∈ V
3837elsuc 6436 . . . . . 6 ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
3936, 38sylib 217 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
4039orcomd 869 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41 fvex 6904 . . . . . . . 8 (𝑅1𝐴) ∈ V
42 elpwi 4605 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 ⊆ (𝑅1𝐴))
4342ad2antlr 725 . . . . . . . 8 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1𝐴))
44 ssdomg 9021 . . . . . . . 8 ((𝑅1𝐴) ∈ V → (𝑥 ⊆ (𝑅1𝐴) → 𝑥 ≼ (𝑅1𝐴)))
4541, 43, 44mpsyl 68 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≼ (𝑅1𝐴))
46 rankcf 10809 . . . . . . . . . 10 ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47 fveq2 6891 . . . . . . . . . . . 12 ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48 elina 10719 . . . . . . . . . . . . 13 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
4948simp2bi 1143 . . . . . . . . . . . 12 (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
5047, 49sylan9eqr 2788 . . . . . . . . . . 11 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
5150breq2d 5156 . . . . . . . . . 10 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥𝐴))
5246, 51mtbii 325 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥𝐴)
53 inar1 10807 . . . . . . . . . . 11 (𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)
54 sdomentr 9139 . . . . . . . . . . . 12 ((𝑥 ≺ (𝑅1𝐴) ∧ (𝑅1𝐴) ≈ 𝐴) → 𝑥𝐴)
5554expcom 412 . . . . . . . . . . 11 ((𝑅1𝐴) ≈ 𝐴 → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5653, 55syl 17 . . . . . . . . . 10 (𝐴 ∈ Inacc → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5756adantr 479 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5852, 57mtod 197 . . . . . . . 8 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
5958adantlr 713 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
60 bren2 9004 . . . . . . 7 (𝑥 ≈ (𝑅1𝐴) ↔ (𝑥 ≼ (𝑅1𝐴) ∧ ¬ 𝑥 ≺ (𝑅1𝐴)))
6145, 59, 60sylanbrc 581 . . . . . 6 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅1𝐴))
6261ex 411 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) = 𝐴𝑥 ≈ (𝑅1𝐴)))
63 r1elwf 9830 . . . . . . . . 9 (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 (𝑅1 “ On))
6433, 63biimtrrdi 253 . . . . . . . 8 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 (𝑅1 “ On)))
6564imp 405 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
66 r1fnon 9801 . . . . . . . . . 10 𝑅1 Fn On
6766fndmi 6654 . . . . . . . . 9 dom 𝑅1 = On
6830, 67eleqtrrdi 2837 . . . . . . . 8 (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
6968adantr 479 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
70 rankr1ag 9836 . . . . . . 7 ((𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7165, 69, 70syl2anc 582 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7271biimprd 247 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) ∈ 𝐴𝑥 ∈ (𝑅1𝐴)))
7362, 72orim12d 962 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴) → (𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴))))
7440, 73mpd 15 . . 3 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))
7574ralrimiva 3136 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))
76 eltsk2g 10783 . . 3 ((𝑅1𝐴) ∈ V → ((𝑅1𝐴) ∈ Tarski ↔ (∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)) ∧ ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))))
7741, 76ax-mp 5 . 2 ((𝑅1𝐴) ∈ Tarski ↔ (∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)) ∧ ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴))))
7829, 75, 77sylanbrc 581 1 (𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3463  wss 3947  c0 4323  𝒫 cpw 4598   cuni 4906   ciun 4994   class class class wbr 5144  dom cdm 5673  cima 5676  Oncon0 6366  Lim wlim 6367  suc csuc 6368  cfv 6544  cen 8961  cdom 8962  csdm 8963  𝑅1cr1 9796  rankcrnk 9797  cfccf 9971  Inaccwcwina 10714  Inacccina 10715  Tarskictsk 10780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-inf2 9675  ax-ac2 10495
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-iin 4997  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9544  df-r1 9798  df-rank 9799  df-card 9973  df-cf 9975  df-acn 9976  df-ac 10150  df-wina 10716  df-ina 10717  df-tsk 10781
This theorem is referenced by:  r1omtsk  10811  r1tskina  10814  grutsk  10854  inagrud  44005
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