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Theorem hsmexlem9 10426
Description: Lemma for hsmex 10433. 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 7890 . 2 (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏))
2 fveq2 6891 . . . 4 (𝑎 = ∅ → (𝐻𝑎) = (𝐻‘∅))
3 hsmexlem7.h . . . . . 6 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
43hsmexlem7 10424 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
5 harcl 9560 . . . . 5 (har‘𝒫 𝑋) ∈ On
64, 5eqeltri 2828 . . . 4 (𝐻‘∅) ∈ On
72, 6eqeltrdi 2840 . . 3 (𝑎 = ∅ → (𝐻𝑎) ∈ On)
83hsmexlem8 10425 . . . . . 6 (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻𝑏))))
9 harcl 9560 . . . . . 6 (har‘𝒫 (𝑋 × (𝐻𝑏))) ∈ On
108, 9eqeltrdi 2840 . . . . 5 (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On)
11 fveq2 6891 . . . . . 6 (𝑎 = suc 𝑏 → (𝐻𝑎) = (𝐻‘suc 𝑏))
1211eleq1d 2817 . . . . 5 (𝑎 = suc 𝑏 → ((𝐻𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On))
1310, 12syl5ibrcom 246 . . . 4 (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On))
1413rexlimiv 3147 . . 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 1540  wcel 2105  wrex 3069  Vcvv 3473  c0 4322  𝒫 cpw 4602  cmpt 5231   × cxp 5674  cres 5678  Oncon0 6364  suc csuc 6366  cfv 6543  ωcom 7859  reccrdg 8415  harchar 9557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-en 8946  df-dom 8947  df-oi 9511  df-har 9558
This theorem is referenced by:  hsmexlem4  10430  hsmexlem5  10431
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