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Theorem r1fin 9195
Description: The first ω levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.)
Assertion
Ref Expression
r1fin (𝐴 ∈ ω → (𝑅1𝐴) ∈ Fin)

Proof of Theorem r1fin
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6659 . . 3 (𝑛 = ∅ → (𝑅1𝑛) = (𝑅1‘∅))
21eleq1d 2900 . 2 (𝑛 = ∅ → ((𝑅1𝑛) ∈ Fin ↔ (𝑅1‘∅) ∈ Fin))
3 fveq2 6659 . . 3 (𝑛 = 𝑚 → (𝑅1𝑛) = (𝑅1𝑚))
43eleq1d 2900 . 2 (𝑛 = 𝑚 → ((𝑅1𝑛) ∈ Fin ↔ (𝑅1𝑚) ∈ Fin))
5 fveq2 6659 . . 3 (𝑛 = suc 𝑚 → (𝑅1𝑛) = (𝑅1‘suc 𝑚))
65eleq1d 2900 . 2 (𝑛 = suc 𝑚 → ((𝑅1𝑛) ∈ Fin ↔ (𝑅1‘suc 𝑚) ∈ Fin))
7 fveq2 6659 . . 3 (𝑛 = 𝐴 → (𝑅1𝑛) = (𝑅1𝐴))
87eleq1d 2900 . 2 (𝑛 = 𝐴 → ((𝑅1𝑛) ∈ Fin ↔ (𝑅1𝐴) ∈ Fin))
9 r10 9190 . . 3 (𝑅1‘∅) = ∅
10 0fin 8739 . . 3 ∅ ∈ Fin
119, 10eqeltri 2912 . 2 (𝑅1‘∅) ∈ Fin
12 r1funlim 9188 . . . . . . . . 9 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1312simpri 489 . . . . . . . 8 Lim dom 𝑅1
14 limomss 7576 . . . . . . . 8 (Lim dom 𝑅1 → ω ⊆ dom 𝑅1)
1513, 14ax-mp 5 . . . . . . 7 ω ⊆ dom 𝑅1
1615sseli 3949 . . . . . 6 (𝑚 ∈ ω → 𝑚 ∈ dom 𝑅1)
17 r1sucg 9191 . . . . . 6 (𝑚 ∈ dom 𝑅1 → (𝑅1‘suc 𝑚) = 𝒫 (𝑅1𝑚))
1816, 17syl 17 . . . . 5 (𝑚 ∈ ω → (𝑅1‘suc 𝑚) = 𝒫 (𝑅1𝑚))
1918eleq1d 2900 . . . 4 (𝑚 ∈ ω → ((𝑅1‘suc 𝑚) ∈ Fin ↔ 𝒫 (𝑅1𝑚) ∈ Fin))
20 pwfi 8812 . . . 4 ((𝑅1𝑚) ∈ Fin ↔ 𝒫 (𝑅1𝑚) ∈ Fin)
2119, 20syl6rbbr 293 . . 3 (𝑚 ∈ ω → ((𝑅1𝑚) ∈ Fin ↔ (𝑅1‘suc 𝑚) ∈ Fin))
2221biimpd 232 . 2 (𝑚 ∈ ω → ((𝑅1𝑚) ∈ Fin → (𝑅1‘suc 𝑚) ∈ Fin))
232, 4, 6, 8, 11, 22finds 7600 1 (𝐴 ∈ ω → (𝑅1𝐴) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wss 3919  c0 4276  𝒫 cpw 4522  dom cdm 5543  Lim wlim 6180  suc csuc 6181  Fun wfun 6338  cfv 6344  ωcom 7571  Fincfn 8501  𝑅1cr1 9184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-r1 9186
This theorem is referenced by:  ackbij2lem2  9656  ackbij2  9659
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