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Theorem tskr1om 9781
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 8708.) (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 6352 . . . . . . 7 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
21eleq1d 2824 . . . . . 6 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
3 fveq2 6352 . . . . . . 7 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43eleq1d 2824 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1𝑦) ∈ 𝑇))
5 fveq2 6352 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
65eleq1d 2824 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
7 r10 8804 . . . . . . 7 (𝑅1‘∅) = ∅
8 tsk0 9777 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
97, 8syl5eqel 2843 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
10 tskpw 9767 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → 𝒫 (𝑅1𝑦) ∈ 𝑇)
11 nnon 7236 . . . . . . . . . . 11 (𝑦 ∈ ω → 𝑦 ∈ On)
12 r1suc 8806 . . . . . . . . . . 11 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1311, 12syl 17 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1413eleq1d 2824 . . . . . . . . 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 451 . . . . . . 7 (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
1716adantrd 485 . . . . . 6 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
182, 4, 6, 9, 17finds2 7259 . . . . 5 (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇))
19 eleq1 2827 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑇𝑦𝑇))
2019imbi2d 329 . . . . 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 3165 . . 3 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇))
23 r1fnon 8803 . . . . 5 𝑅1 Fn On
24 fnfun 6149 . . . . 5 (𝑅1 Fn On → Fun 𝑅1)
2523, 24ax-mp 5 . . . 4 Fun 𝑅1
26 fvelima 6410 . . . 4 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2725, 26mpan 708 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2822, 27syl11 33 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑇))
2928ssrdv 3750 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wne 2932  wrex 3051  wss 3715  c0 4058  𝒫 cpw 4302  cima 5269  Oncon0 5884  suc csuc 5886  Fun wfun 6043   Fn wfn 6044  cfv 6049  ωcom 7230  𝑅1cr1 8798  Tarskictsk 9762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800  df-tsk 9763
This theorem is referenced by:  tskr1om2  9782  tskinf  9783
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