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Theorem hsmexlem9 10462
Description: Lemma for hsmex 10469. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Hypothesis
Ref Expression
hsmexlem7.h 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
Assertion
Ref Expression
hsmexlem9 (𝑎 ∈ ω → (𝐻𝑎) ∈ On)
Distinct variable groups:   𝑧,𝑋   𝑧,𝑎
Allowed substitution hints:   𝐻(𝑧,𝑎)   𝑋(𝑎)

Proof of Theorem hsmexlem9
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 nn0suc 7916 . 2 (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏))
2 fveq2 6906 . . . 4 (𝑎 = ∅ → (𝐻𝑎) = (𝐻‘∅))
3 hsmexlem7.h . . . . . 6 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
43hsmexlem7 10460 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
5 harcl 9596 . . . . 5 (har‘𝒫 𝑋) ∈ On
64, 5eqeltri 2834 . . . 4 (𝐻‘∅) ∈ On
72, 6eqeltrdi 2846 . . 3 (𝑎 = ∅ → (𝐻𝑎) ∈ On)
83hsmexlem8 10461 . . . . . 6 (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻𝑏))))
9 harcl 9596 . . . . . 6 (har‘𝒫 (𝑋 × (𝐻𝑏))) ∈ On
108, 9eqeltrdi 2846 . . . . 5 (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On)
11 fveq2 6906 . . . . . 6 (𝑎 = suc 𝑏 → (𝐻𝑎) = (𝐻‘suc 𝑏))
1211eleq1d 2823 . . . . 5 (𝑎 = suc 𝑏 → ((𝐻𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On))
1310, 12syl5ibrcom 247 . . . 4 (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On))
1413rexlimiv 3145 . . 3 (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On)
157, 14jaoi 857 . 2 ((𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏) → (𝐻𝑎) ∈ On)
161, 15syl 17 1 (𝑎 ∈ ω → (𝐻𝑎) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  c0 4338  𝒫 cpw 4604  cmpt 5230   × cxp 5686  cres 5690  Oncon0 6385  suc csuc 6387  cfv 6562  ωcom 7886  reccrdg 8447  harchar 9593
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-rmo 3377  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-se 5641  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-isom 6571  df-riota 7387  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-en 8984  df-dom 8985  df-oi 9547  df-har 9594
This theorem is referenced by:  hsmexlem4  10466  hsmexlem5  10467
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