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Theorem r1tskina 10726
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 2941 . . . . 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 9774 . . . . . . . . . . . . . . . 16 On βŠ† βˆͺ (𝑅1 β€œ On)
54sseli 3944 . . . . . . . . . . . . . . 15 (𝐴 ∈ On β†’ 𝐴 ∈ βˆͺ (𝑅1 β€œ On))
6 eqid 2733 . . . . . . . . . . . . . . . 16 (rankβ€˜π΄) = (rankβ€˜π΄)
7 rankr1c 9765 . . . . . . . . . . . . . . . 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 9711 . . . . . . . . . . . . . . . . 17 𝑅1 Fn On
1211fndmi 6610 . . . . . . . . . . . . . . . 16 dom 𝑅1 = On
1312eleq2i 2826 . . . . . . . . . . . . . . 15 (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
14 rankonid 9773 . . . . . . . . . . . . . . 15 (𝐴 ∈ dom 𝑅1 ↔ (rankβ€˜π΄) = 𝐴)
1513, 14bitr3i 277 . . . . . . . . . . . . . 14 (𝐴 ∈ On ↔ (rankβ€˜π΄) = 𝐴)
16 fveq2 6846 . . . . . . . . . . . . . 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 9775 . . . . . . . . . . . . . 14 (𝐴 ∈ dom 𝑅1 β†’ 𝐴 βŠ† (𝑅1β€˜π΄))
2113, 20sylbir 234 . . . . . . . . . . . . 13 (𝐴 ∈ On β†’ 𝐴 βŠ† (𝑅1β€˜π΄))
22 tsken 10698 . . . . . . . . . . . . 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 9911 . . . . . . . . 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 3948 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ (𝑅1β€˜π΄))
33 tsksdom 10700 . . . . . . . . . . . . 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 8951 . . . . . . . . . . . . 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 9061 . . . . . . . . . . . 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 3140 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑅1β€˜π΄) ∈ Tarski) β†’ βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰Ί 𝐴)
41 iscard 9919 . . . . . . . . . 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 9712 . . . . . . . . . . 11 (𝑅1β€˜βˆ…) = βˆ…
46 on0eln0 6377 . . . . . . . . . . . . 13 (𝐴 ∈ On β†’ (βˆ… ∈ 𝐴 ↔ 𝐴 β‰  βˆ…))
4746biimpar 479 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐴 β‰  βˆ…) β†’ βˆ… ∈ 𝐴)
48 r1sdom 9718 . . . . . . . . . . . 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 5145 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐴 β‰  βˆ…) β†’ βˆ… β‰Ί (𝑅1β€˜π΄))
51 fvex 6859 . . . . . . . . . . 11 (𝑅1β€˜π΄) ∈ V
52510sdom 9057 . . . . . . . . . 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 10725 . . . . . . . 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 6846 . . . . 5 (𝐴 = βˆ… β†’ (𝑅1β€˜π΄) = (𝑅1β€˜βˆ…))
6362, 45eqtrdi 2789 . . . 4 (𝐴 = βˆ… β†’ (𝑅1β€˜π΄) = βˆ…)
64 0tsk 10699 . . . 4 βˆ… ∈ Tarski
6563, 64eqeltrdi 2842 . . 3 (𝐴 = βˆ… β†’ (𝑅1β€˜π΄) ∈ Tarski)
66 inatsk 10722 . . 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 2940  βˆ€wral 3061   βŠ† wss 3914  βˆ…c0 4286  βˆͺ cuni 4869   class class class wbr 5109  dom cdm 5637   β€œ cima 5640  Oncon0 6321  suc csuc 6323  β€˜cfv 6500   β‰ˆ cen 8886   β‰Ί csdm 8888  π‘…1cr1 9706  rankcrnk 9707  cardccrd 9879  Inacccina 10627  Tarskictsk 10692
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-ac2 10407
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-smo 8296  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-map 8773  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-oi 9454  df-har 9501  df-r1 9708  df-rank 9709  df-card 9883  df-aleph 9884  df-cf 9885  df-acn 9886  df-ac 10060  df-wina 10628  df-ina 10629  df-tsk 10693
This theorem is referenced by: (None)
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