MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tskr1om Structured version   Visualization version   GIF version

Theorem tskr1om 10804
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 9675.) (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 6906 . . . . . . 7 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
21eleq1d 2823 . . . . . 6 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
3 fveq2 6906 . . . . . . 7 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43eleq1d 2823 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1𝑦) ∈ 𝑇))
5 fveq2 6906 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
65eleq1d 2823 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
7 r10 9805 . . . . . . 7 (𝑅1‘∅) = ∅
8 tsk0 10800 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
97, 8eqeltrid 2842 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
10 tskpw 10790 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → 𝒫 (𝑅1𝑦) ∈ 𝑇)
11 nnon 7892 . . . . . . . . . . 11 (𝑦 ∈ ω → 𝑦 ∈ On)
12 r1suc 9807 . . . . . . . . . . 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, 14imbitrrid 246 . . . . . . . 8 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ (𝑅1𝑦) ∈ 𝑇) → (𝑅1‘suc 𝑦) ∈ 𝑇))
1615expd 415 . . . . . . 7 (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
1716adantrd 491 . . . . . 6 (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
182, 4, 6, 9, 17finds2 7920 . . . . 5 (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇))
19 eleq1 2826 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑇𝑦𝑇))
2019imbi2d 340 . . . . 5 ((𝑅1𝑥) = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1𝑥) ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇)))
2118, 20syl5ibcom 245 . . . 4 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇)))
2221rexlimiv 3145 . . 3 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦𝑇))
23 r1fnon 9804 . . . . 5 𝑅1 Fn On
24 fnfun 6668 . . . . 5 (𝑅1 Fn On → Fun 𝑅1)
2523, 24ax-mp 5 . . . 4 Fun 𝑅1
26 fvelima 6973 . . . 4 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2725, 26mpan 690 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2822, 27syl11 33 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑇))
2928ssrdv 4000 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  wrex 3067  wss 3962  c0 4338  𝒫 cpw 4604  cima 5691  Oncon0 6385  suc csuc 6387  Fun wfun 6556   Fn wfn 6557  cfv 6562  ωcom 7886  𝑅1cr1 9799  Tarskictsk 10785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-r1 9801  df-tsk 10786
This theorem is referenced by:  tskr1om2  10805  tskinf  10806
  Copyright terms: Public domain W3C validator