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Theorem tskr1om 9468
 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 8418.) (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 r1fnon 8513 . . . . . 6 𝑅1 Fn On
2 fnfun 5902 . . . . . 6 (𝑅1 Fn On → Fun 𝑅1)
31, 2ax-mp 5 . . . . 5 Fun 𝑅1
4 fvelima 6158 . . . . 5 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
53, 4mpan 702 . . . 4 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
6 fveq2 6103 . . . . . . . 8 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
76eleq1d 2672 . . . . . . 7 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
8 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
98eleq1d 2672 . . . . . . 7 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1𝑦) ∈ 𝑇))
10 fveq2 6103 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1110eleq1d 2672 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
12 r10 8514 . . . . . . . 8 (𝑅1‘∅) = ∅
13 tsk0 9464 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
1412, 13syl5eqel 2692 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
15 tskpw 9454 . . . . . . . . . 10 ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → 𝒫 (𝑅1𝑦) ∈ 𝑇)
16 nnon 6963 . . . . . . . . . . . 12 (𝑦 ∈ ω → 𝑦 ∈ On)
17 r1suc 8516 . . . . . . . . . . . 12 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1816, 17syl 17 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1918eleq1d 2672 . . . . . . . . . 10 (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑇 ↔ 𝒫 (𝑅1𝑦) ∈ 𝑇))
2015, 19syl5ibr 235 . . . . . . . . 9 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → (𝑅1‘suc 𝑦) ∈ 𝑇))
2120expd 451 . . . . . . . 8 (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
2221adantrd 483 . . . . . . 7 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
237, 9, 11, 14, 22finds2 6986 . . . . . 6 (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇))
24 eleq1 2676 . . . . . . 7 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑇𝑦𝑇))
2524imbi2d 329 . . . . . 6 ((𝑅1𝑥) = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇)))
2623, 25syl5ibcom 234 . . . . 5 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇)))
2726rexlimiv 3009 . . . 4 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇))
285, 27syl 17 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇))
2928com12 32 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑇))
3029ssrdv 3574 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   “ cima 5041  Oncon0 5640  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  ωcom 6957  𝑅1cr1 8508  Tarskictsk 9449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-tsk 9450 This theorem is referenced by:  tskr1om2  9469  tskinf  9470
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