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Theorem tskr1om 9533
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 8479.) (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 6148 . . . . . . 7 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
21eleq1d 2683 . . . . . 6 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
3 fveq2 6148 . . . . . . 7 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43eleq1d 2683 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1𝑦) ∈ 𝑇))
5 fveq2 6148 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
65eleq1d 2683 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
7 r10 8575 . . . . . . 7 (𝑅1‘∅) = ∅
8 tsk0 9529 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
97, 8syl5eqel 2702 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
10 tskpw 9519 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → 𝒫 (𝑅1𝑦) ∈ 𝑇)
11 nnon 7018 . . . . . . . . . . 11 (𝑦 ∈ ω → 𝑦 ∈ On)
12 r1suc 8577 . . . . . . . . . . 11 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1311, 12syl 17 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1413eleq1d 2683 . . . . . . . . 9 (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑇 ↔ 𝒫 (𝑅1𝑦) ∈ 𝑇))
1510, 14syl5ibr 236 . . . . . . . 8 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → (𝑅1‘suc 𝑦) ∈ 𝑇))
1615expd 452 . . . . . . 7 (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
1716adantrd 484 . . . . . 6 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
182, 4, 6, 9, 17finds2 7041 . . . . 5 (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇))
19 eleq1 2686 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑇𝑦𝑇))
2019imbi2d 330 . . . . 5 ((𝑅1𝑥) = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇)))
2118, 20syl5ibcom 235 . . . 4 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇)))
2221rexlimiv 3020 . . 3 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇))
23 r1fnon 8574 . . . . 5 𝑅1 Fn On
24 fnfun 5946 . . . . 5 (𝑅1 Fn On → Fun 𝑅1)
2523, 24ax-mp 5 . . . 4 Fun 𝑅1
26 fvelima 6205 . . . 4 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2725, 26mpan 705 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2822, 27syl11 33 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑇))
2928ssrdv 3589 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  wss 3555  c0 3891  𝒫 cpw 4130  cima 5077  Oncon0 5682  suc csuc 5684  Fun wfun 5841   Fn wfn 5842  cfv 5847  ωcom 7012  𝑅1cr1 8569  Tarskictsk 9514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-r1 8571  df-tsk 9515
This theorem is referenced by:  tskr1om2  9534  tskinf  9535
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