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Theorem hsmexlem9 9836
 Description: Lemma for hsmex 9843. 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 7586 . 2 (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏))
2 fveq2 6645 . . . 4 (𝑎 = ∅ → (𝐻𝑎) = (𝐻‘∅))
3 hsmexlem7.h . . . . . 6 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
43hsmexlem7 9834 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
5 harcl 9007 . . . . 5 (har‘𝒫 𝑋) ∈ On
64, 5eqeltri 2886 . . . 4 (𝐻‘∅) ∈ On
72, 6eqeltrdi 2898 . . 3 (𝑎 = ∅ → (𝐻𝑎) ∈ On)
83hsmexlem8 9835 . . . . . 6 (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻𝑏))))
9 harcl 9007 . . . . . 6 (har‘𝒫 (𝑋 × (𝐻𝑏))) ∈ On
108, 9eqeltrdi 2898 . . . . 5 (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On)
11 fveq2 6645 . . . . . 6 (𝑎 = suc 𝑏 → (𝐻𝑎) = (𝐻‘suc 𝑏))
1211eleq1d 2874 . . . . 5 (𝑎 = suc 𝑏 → ((𝐻𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On))
1310, 12syl5ibrcom 250 . . . 4 (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On))
1413rexlimiv 3239 . . 3 (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On)
157, 14jaoi 854 . 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 844   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441  ∅c0 4243  𝒫 cpw 4497   ↦ cmpt 5110   × cxp 5517   ↾ cres 5521  Oncon0 6159  suc csuc 6161  ‘cfv 6324  ωcom 7560  reccrdg 8028  harchar 9004 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-en 8493  df-dom 8494  df-oi 8958  df-har 9005 This theorem is referenced by:  hsmexlem4  9840  hsmexlem5  9841
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