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Theorem inatsk 9937
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 9849 . . . . . 6 (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2 winaon 9847 . . . . . . . . . 10 (𝐴 ∈ Inaccw𝐴 ∈ On)
3 winalim 9854 . . . . . . . . . 10 (𝐴 ∈ Inaccw → Lim 𝐴)
4 r1lim 8934 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
52, 3, 4syl2anc 579 . . . . . . . . 9 (𝐴 ∈ Inaccw → (𝑅1𝐴) = 𝑦𝐴 (𝑅1𝑦))
65eleq2d 2845 . . . . . . . 8 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ 𝑥 𝑦𝐴 (𝑅1𝑦)))
7 eliun 4759 . . . . . . . 8 (𝑥 𝑦𝐴 (𝑅1𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦))
86, 7syl6bb 279 . . . . . . 7 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) ↔ ∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦)))
9 onelon 6003 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
102, 9sylan 575 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → 𝑦 ∈ On)
11 r1pw 9007 . . . . . . . . . 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 7329 . . . . . . . . . . . . 13 (Lim 𝐴 → (𝑦𝐴 ↔ suc 𝑦𝐴))
143, 13syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (𝑦𝐴 ↔ suc 𝑦𝐴))
15 r1ord2 8943 . . . . . . . . . . . . 13 (𝐴 ∈ On → (suc 𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
162, 15syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Inaccw → (suc 𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
1714, 16sylbid 232 . . . . . . . . . . 11 (𝐴 ∈ Inaccw → (𝑦𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴)))
1817imp 397 . . . . . . . . . 10 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝐴))
1918sseld 3820 . . . . . . . . 9 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2012, 19sylbid 232 . . . . . . . 8 ((𝐴 ∈ Inaccw𝑦𝐴) → (𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2120rexlimdva 3213 . . . . . . 7 (𝐴 ∈ Inaccw → (∃𝑦𝐴 𝑥 ∈ (𝑅1𝑦) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
228, 21sylbid 232 . . . . . 6 (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
231, 22syl 17 . . . . 5 (𝐴 ∈ Inacc → (𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ∈ (𝑅1𝐴)))
2423imp 397 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → 𝒫 𝑥 ∈ (𝑅1𝐴))
25 elssuni 4704 . . . . 5 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 (𝑅1𝐴))
26 r1tr2 8939 . . . . 5 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2725, 26syl6ss 3833 . . . 4 (𝒫 𝑥 ∈ (𝑅1𝐴) → 𝒫 𝑥 ⊆ (𝑅1𝐴))
2824, 27jccil 518 . . 3 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1𝐴)) → (𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
2928ralrimiva 3148 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ (𝑅1𝐴)(𝒫 𝑥 ⊆ (𝑅1𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1𝐴)))
301, 2syl 17 . . . . . . . . 9 (𝐴 ∈ Inacc → 𝐴 ∈ On)
31 r1suc 8932 . . . . . . . . . 10 (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
3231eleq2d 2845 . . . . . . . . 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 8960 . . . . . . . 8 (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
3533, 34syl6bir 246 . . . . . . 7 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → (rank‘𝑥) ∈ suc 𝐴))
3635imp 397 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37 fvex 6461 . . . . . . 7 (rank‘𝑥) ∈ V
3837elsuc 6047 . . . . . 6 ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
3936, 38sylib 210 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
4039orcomd 860 . . . 4 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41 fvex 6461 . . . . . . . 8 (𝑅1𝐴) ∈ V
42 elpwi 4389 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 ⊆ (𝑅1𝐴))
4342ad2antlr 717 . . . . . . . 8 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1𝐴))
44 ssdomg 8289 . . . . . . . 8 ((𝑅1𝐴) ∈ V → (𝑥 ⊆ (𝑅1𝐴) → 𝑥 ≼ (𝑅1𝐴)))
4541, 43, 44mpsyl 68 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≼ (𝑅1𝐴))
46 rankcf 9936 . . . . . . . . . 10 ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47 fveq2 6448 . . . . . . . . . . . 12 ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48 elina 9846 . . . . . . . . . . . . 13 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
4948simp2bi 1137 . . . . . . . . . . . 12 (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
5047, 49sylan9eqr 2836 . . . . . . . . . . 11 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
5150breq2d 4900 . . . . . . . . . 10 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥𝐴))
5246, 51mtbii 318 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥𝐴)
53 inar1 9934 . . . . . . . . . . 11 (𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)
54 sdomentr 8384 . . . . . . . . . . . 12 ((𝑥 ≺ (𝑅1𝐴) ∧ (𝑅1𝐴) ≈ 𝐴) → 𝑥𝐴)
5554expcom 404 . . . . . . . . . . 11 ((𝑅1𝐴) ≈ 𝐴 → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5653, 55syl 17 . . . . . . . . . 10 (𝐴 ∈ Inacc → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5756adantr 474 . . . . . . . . 9 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (𝑅1𝐴) → 𝑥𝐴))
5852, 57mtod 190 . . . . . . . 8 ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
5958adantlr 705 . . . . . . 7 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1𝐴))
60 bren2 8274 . . . . . . 7 (𝑥 ≈ (𝑅1𝐴) ↔ (𝑥 ≼ (𝑅1𝐴) ∧ ¬ 𝑥 ≺ (𝑅1𝐴)))
6145, 59, 60sylanbrc 578 . . . . . 6 (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅1𝐴))
6261ex 403 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) = 𝐴𝑥 ≈ (𝑅1𝐴)))
63 r1elwf 8958 . . . . . . . . 9 (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 (𝑅1 “ On))
6433, 63syl6bir 246 . . . . . . . 8 (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1𝐴) → 𝑥 (𝑅1 “ On)))
6564imp 397 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝑥 (𝑅1 “ On))
66 r1fnon 8929 . . . . . . . . . 10 𝑅1 Fn On
67 fndm 6237 . . . . . . . . . 10 (𝑅1 Fn On → dom 𝑅1 = On)
6866, 67ax-mp 5 . . . . . . . . 9 dom 𝑅1 = On
6930, 68syl6eleqr 2870 . . . . . . . 8 (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
7069adantr 474 . . . . . . 7 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → 𝐴 ∈ dom 𝑅1)
71 rankr1ag 8964 . . . . . . 7 ((𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7265, 70, 71syl2anc 579 . . . . . 6 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
7372biimprd 240 . . . . 5 ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1𝐴)) → ((rank‘𝑥) ∈ 𝐴𝑥 ∈ (𝑅1𝐴)))
7462, 73orim12d 950 . . . 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 3148 . 2 (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1𝐴)(𝑥 ≈ (𝑅1𝐴) ∨ 𝑥 ∈ (𝑅1𝐴)))
77 eltsk2g 9910 . . 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 578 1 (𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 836   = wceq 1601  wcel 2107  wne 2969  wral 3090  wrex 3091  Vcvv 3398  wss 3792  c0 4141  𝒫 cpw 4379   cuni 4673   ciun 4755   class class class wbr 4888  dom cdm 5357  cima 5360  Oncon0 5978  Lim wlim 5979  suc csuc 5980   Fn wfn 6132  cfv 6137  cen 8240  cdom 8241  csdm 8242  𝑅1cr1 8924  rankcrnk 8925  cfccf 9098  Inaccwcwina 9841  Inacccina 9842  Tarskictsk 9907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-ac2 9622
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-oi 8706  df-r1 8926  df-rank 8927  df-card 9100  df-cf 9102  df-acn 9103  df-ac 9274  df-wina 9843  df-ina 9844  df-tsk 9908
This theorem is referenced by:  r1omtsk  9938  r1tskina  9941  grutsk  9981
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