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Theorem tskr1om 10523
Description: A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 9396.) (Contributed by Mario Carneiro, 24-Jun-2013.)
Assertion
Ref Expression
tskr1om ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Proof of Theorem tskr1om
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6774 . . . . . . 7 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
21eleq1d 2823 . . . . . 6 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
3 fveq2 6774 . . . . . . 7 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43eleq1d 2823 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1𝑦) ∈ 𝑇))
5 fveq2 6774 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
65eleq1d 2823 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
7 r10 9526 . . . . . . 7 (𝑅1‘∅) = ∅
8 tsk0 10519 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
97, 8eqeltrid 2843 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
10 tskpw 10509 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → 𝒫 (𝑅1𝑦) ∈ 𝑇)
11 nnon 7718 . . . . . . . . . . 11 (𝑦 ∈ ω → 𝑦 ∈ On)
12 r1suc 9528 . . . . . . . . . . 11 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1311, 12syl 17 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1413eleq1d 2823 . . . . . . . . 9 (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑇 ↔ 𝒫 (𝑅1𝑦) ∈ 𝑇))
1510, 14syl5ibr 245 . . . . . . . 8 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → (𝑅1‘suc 𝑦) ∈ 𝑇))
1615expd 416 . . . . . . 7 (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
1716adantrd 492 . . . . . 6 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
182, 4, 6, 9, 17finds2 7747 . . . . 5 (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇))
19 eleq1 2826 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑇𝑦𝑇))
2019imbi2d 341 . . . . 5 ((𝑅1𝑥) = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇)))
2118, 20syl5ibcom 244 . . . 4 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇)))
2221rexlimiv 3209 . . 3 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇))
23 r1fnon 9525 . . . . 5 𝑅1 Fn On
24 fnfun 6533 . . . . 5 (𝑅1 Fn On → Fun 𝑅1)
2523, 24ax-mp 5 . . . 4 Fun 𝑅1
26 fvelima 6835 . . . 4 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2725, 26mpan 687 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2822, 27syl11 33 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑇))
2928ssrdv 3927 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  wrex 3065  wss 3887  c0 4256  𝒫 cpw 4533  cima 5592  Oncon0 6266  suc csuc 6268  Fun wfun 6427   Fn wfn 6428  cfv 6433  ωcom 7712  𝑅1cr1 9520  Tarskictsk 10504
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-tsk 10505
This theorem is referenced by:  tskr1om2  10524  tskinf  10525
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