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Theorem r1tskina 10777
Description: There is a direct relationship between transitive Tarski classes and inaccessible cardinals: the Tarski classes that occur in the cumulative hierarchy are exactly at the strongly inaccessible cardinals. (Contributed by Mario Carneiro, 8-Jun-2013.)
Assertion
Ref Expression
r1tskina (𝐴 ∈ On β†’ ((𝑅1β€˜π΄) ∈ Tarski ↔ (𝐴 = βˆ… ∨ 𝐴 ∈ Inacc)))

Proof of Theorem r1tskina
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 df-ne 2942 . . . . 5 (𝐴 β‰  βˆ… ↔ Β¬ 𝐴 = βˆ…)
2 simplr 768 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ (𝑅1β€˜π΄) ∈ Tarski)
3 simpll 766 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ 𝐴 ∈ On)
4 onwf 9825 . . . . . . . . . . . . . . . 16 On βŠ† βˆͺ (𝑅1 β€œ On)
54sseli 3979 . . . . . . . . . . . . . . 15 (𝐴 ∈ On β†’ 𝐴 ∈ βˆͺ (𝑅1 β€œ On))
6 eqid 2733 . . . . . . . . . . . . . . . 16 (rankβ€˜π΄) = (rankβ€˜π΄)
7 rankr1c 9816 . . . . . . . . . . . . . . . 16 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ ((rankβ€˜π΄) = (rankβ€˜π΄) ↔ (Β¬ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄)) ∧ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄)))))
86, 7mpbii 232 . . . . . . . . . . . . . . 15 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ (Β¬ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄)) ∧ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄))))
95, 8syl 17 . . . . . . . . . . . . . 14 (𝐴 ∈ On β†’ (Β¬ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄)) ∧ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄))))
109simpld 496 . . . . . . . . . . . . 13 (𝐴 ∈ On β†’ Β¬ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄)))
11 r1fnon 9762 . . . . . . . . . . . . . . . . 17 𝑅1 Fn On
1211fndmi 6654 . . . . . . . . . . . . . . . 16 dom 𝑅1 = On
1312eleq2i 2826 . . . . . . . . . . . . . . 15 (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
14 rankonid 9824 . . . . . . . . . . . . . . 15 (𝐴 ∈ dom 𝑅1 ↔ (rankβ€˜π΄) = 𝐴)
1513, 14bitr3i 277 . . . . . . . . . . . . . 14 (𝐴 ∈ On ↔ (rankβ€˜π΄) = 𝐴)
16 fveq2 6892 . . . . . . . . . . . . . 14 ((rankβ€˜π΄) = 𝐴 β†’ (𝑅1β€˜(rankβ€˜π΄)) = (𝑅1β€˜π΄))
1715, 16sylbi 216 . . . . . . . . . . . . 13 (𝐴 ∈ On β†’ (𝑅1β€˜(rankβ€˜π΄)) = (𝑅1β€˜π΄))
1810, 17neleqtrd 2856 . . . . . . . . . . . 12 (𝐴 ∈ On β†’ Β¬ 𝐴 ∈ (𝑅1β€˜π΄))
1918adantl 483 . . . . . . . . . . 11 (((𝑅1β€˜π΄) ∈ Tarski ∧ 𝐴 ∈ On) β†’ Β¬ 𝐴 ∈ (𝑅1β€˜π΄))
20 onssr1 9826 . . . . . . . . . . . . . 14 (𝐴 ∈ dom 𝑅1 β†’ 𝐴 βŠ† (𝑅1β€˜π΄))
2113, 20sylbir 234 . . . . . . . . . . . . 13 (𝐴 ∈ On β†’ 𝐴 βŠ† (𝑅1β€˜π΄))
22 tsken 10749 . . . . . . . . . . . . 13 (((𝑅1β€˜π΄) ∈ Tarski ∧ 𝐴 βŠ† (𝑅1β€˜π΄)) β†’ (𝐴 β‰ˆ (𝑅1β€˜π΄) ∨ 𝐴 ∈ (𝑅1β€˜π΄)))
2321, 22sylan2 594 . . . . . . . . . . . 12 (((𝑅1β€˜π΄) ∈ Tarski ∧ 𝐴 ∈ On) β†’ (𝐴 β‰ˆ (𝑅1β€˜π΄) ∨ 𝐴 ∈ (𝑅1β€˜π΄)))
2423ord 863 . . . . . . . . . . 11 (((𝑅1β€˜π΄) ∈ Tarski ∧ 𝐴 ∈ On) β†’ (Β¬ 𝐴 β‰ˆ (𝑅1β€˜π΄) β†’ 𝐴 ∈ (𝑅1β€˜π΄)))
2519, 24mt3d 148 . . . . . . . . . 10 (((𝑅1β€˜π΄) ∈ Tarski ∧ 𝐴 ∈ On) β†’ 𝐴 β‰ˆ (𝑅1β€˜π΄))
262, 3, 25syl2anc 585 . . . . . . . . 9 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ 𝐴 β‰ˆ (𝑅1β€˜π΄))
27 carden2b 9962 . . . . . . . . 9 (𝐴 β‰ˆ (𝑅1β€˜π΄) β†’ (cardβ€˜π΄) = (cardβ€˜(𝑅1β€˜π΄)))
2826, 27syl 17 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ (cardβ€˜π΄) = (cardβ€˜(𝑅1β€˜π΄)))
29 simpl 484 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) β†’ 𝐴 ∈ On)
30 simplr 768 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ π‘₯ ∈ 𝐴) β†’ (𝑅1β€˜π΄) ∈ Tarski)
3121adantr 482 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) β†’ 𝐴 βŠ† (𝑅1β€˜π΄))
3231sselda 3983 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ (𝑅1β€˜π΄))
33 tsksdom 10751 . . . . . . . . . . . . 13 (((𝑅1β€˜π΄) ∈ Tarski ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ π‘₯ β‰Ί (𝑅1β€˜π΄))
3430, 32, 33syl2anc 585 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ β‰Ί (𝑅1β€˜π΄))
35 simpll 766 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ π‘₯ ∈ 𝐴) β†’ 𝐴 ∈ On)
3625ensymd 9001 . . . . . . . . . . . . 13 (((𝑅1β€˜π΄) ∈ Tarski ∧ 𝐴 ∈ On) β†’ (𝑅1β€˜π΄) β‰ˆ 𝐴)
3730, 35, 36syl2anc 585 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ π‘₯ ∈ 𝐴) β†’ (𝑅1β€˜π΄) β‰ˆ 𝐴)
38 sdomentr 9111 . . . . . . . . . . . 12 ((π‘₯ β‰Ί (𝑅1β€˜π΄) ∧ (𝑅1β€˜π΄) β‰ˆ 𝐴) β†’ π‘₯ β‰Ί 𝐴)
3934, 37, 38syl2anc 585 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ β‰Ί 𝐴)
4039ralrimiva 3147 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) β†’ βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰Ί 𝐴)
41 iscard 9970 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰Ί 𝐴))
4229, 40, 41sylanbrc 584 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) β†’ (cardβ€˜π΄) = 𝐴)
4342adantr 482 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ (cardβ€˜π΄) = 𝐴)
4428, 43eqtr3d 2775 . . . . . . 7 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ (cardβ€˜(𝑅1β€˜π΄)) = 𝐴)
45 r10 9763 . . . . . . . . . . 11 (𝑅1β€˜βˆ…) = βˆ…
46 on0eln0 6421 . . . . . . . . . . . . 13 (𝐴 ∈ On β†’ (βˆ… ∈ 𝐴 ↔ 𝐴 β‰  βˆ…))
4746biimpar 479 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐴 β‰  βˆ…) β†’ βˆ… ∈ 𝐴)
48 r1sdom 9769 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ βˆ… ∈ 𝐴) β†’ (𝑅1β€˜βˆ…) β‰Ί (𝑅1β€˜π΄))
4947, 48syldan 592 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐴 β‰  βˆ…) β†’ (𝑅1β€˜βˆ…) β‰Ί (𝑅1β€˜π΄))
5045, 49eqbrtrrid 5185 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐴 β‰  βˆ…) β†’ βˆ… β‰Ί (𝑅1β€˜π΄))
51 fvex 6905 . . . . . . . . . . 11 (𝑅1β€˜π΄) ∈ V
52510sdom 9107 . . . . . . . . . 10 (βˆ… β‰Ί (𝑅1β€˜π΄) ↔ (𝑅1β€˜π΄) β‰  βˆ…)
5350, 52sylib 217 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐴 β‰  βˆ…) β†’ (𝑅1β€˜π΄) β‰  βˆ…)
5453adantlr 714 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ (𝑅1β€˜π΄) β‰  βˆ…)
55 tskcard 10776 . . . . . . . 8 (((𝑅1β€˜π΄) ∈ Tarski ∧ (𝑅1β€˜π΄) β‰  βˆ…) β†’ (cardβ€˜(𝑅1β€˜π΄)) ∈ Inacc)
562, 54, 55syl2anc 585 . . . . . . 7 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ (cardβ€˜(𝑅1β€˜π΄)) ∈ Inacc)
5744, 56eqeltrrd 2835 . . . . . 6 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ 𝐴 β‰  βˆ…) β†’ 𝐴 ∈ Inacc)
5857ex 414 . . . . 5 ((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) β†’ (𝐴 β‰  βˆ… β†’ 𝐴 ∈ Inacc))
591, 58biimtrrid 242 . . . 4 ((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) β†’ (Β¬ 𝐴 = βˆ… β†’ 𝐴 ∈ Inacc))
6059orrd 862 . . 3 ((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) β†’ (𝐴 = βˆ… ∨ 𝐴 ∈ Inacc))
6160ex 414 . 2 (𝐴 ∈ On β†’ ((𝑅1β€˜π΄) ∈ Tarski β†’ (𝐴 = βˆ… ∨ 𝐴 ∈ Inacc)))
62 fveq2 6892 . . . . 5 (𝐴 = βˆ… β†’ (𝑅1β€˜π΄) = (𝑅1β€˜βˆ…))
6362, 45eqtrdi 2789 . . . 4 (𝐴 = βˆ… β†’ (𝑅1β€˜π΄) = βˆ…)
64 0tsk 10750 . . . 4 βˆ… ∈ Tarski
6563, 64eqeltrdi 2842 . . 3 (𝐴 = βˆ… β†’ (𝑅1β€˜π΄) ∈ Tarski)
66 inatsk 10773 . . 3 (𝐴 ∈ Inacc β†’ (𝑅1β€˜π΄) ∈ Tarski)
6765, 66jaoi 856 . 2 ((𝐴 = βˆ… ∨ 𝐴 ∈ Inacc) β†’ (𝑅1β€˜π΄) ∈ Tarski)
6861, 67impbid1 224 1 (𝐴 ∈ On β†’ ((𝑅1β€˜π΄) ∈ Tarski ↔ (𝐴 = βˆ… ∨ 𝐴 ∈ Inacc)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   βŠ† wss 3949  βˆ…c0 4323  βˆͺ cuni 4909   class class class wbr 5149  dom cdm 5677   β€œ cima 5680  Oncon0 6365  suc csuc 6367  β€˜cfv 6544   β‰ˆ cen 8936   β‰Ί csdm 8938  π‘…1cr1 9757  rankcrnk 9758  cardccrd 9930  Inacccina 10678  Tarskictsk 10743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-smo 8346  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-har 9552  df-r1 9759  df-rank 9760  df-card 9934  df-aleph 9935  df-cf 9936  df-acn 9937  df-ac 10111  df-wina 10679  df-ina 10680  df-tsk 10744
This theorem is referenced by: (None)
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