Theorem hsmexlem9 9833

Description: Lemma for hsmex 9840. 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 → (𝐻‘𝑎) ∈ 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.

Step	Hyp	Ref	Expression
1		nn0suc 7592	. 2 ⊢ (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏))
2		fveq2 6656	. . . 4 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) = (𝐻‘∅))
3		hsmexlem7.h	. . . . . 6 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
4	3	hsmexlem7 9831	. . . . 5 ⊢ (𝐻‘∅) = (har‘𝒫 𝑋)
5		harcl 9011	. . . . 5 ⊢ (har‘𝒫 𝑋) ∈ On
6	4, 5	eqeltri 2909	. . . 4 ⊢ (𝐻‘∅) ∈ On
7	2, 6	eqeltrdi 2921	. . 3 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) ∈ On)
8	3	hsmexlem8 9832	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻‘𝑏))))
9		harcl 9011	. . . . . 6 ⊢ (har‘𝒫 (𝑋 × (𝐻‘𝑏))) ∈ On
10	8, 9	eqeltrdi 2921	. . . . 5 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On)
11		fveq2 6656	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → (𝐻‘𝑎) = (𝐻‘suc 𝑏))
12	11	eleq1d 2897	. . . . 5 ⊢ (𝑎 = suc 𝑏 → ((𝐻‘𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On))
13	10, 12	syl5ibrcom 249	. . . 4 ⊢ (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On))
14	13	rexlimiv 3280	. . 3 ⊢ (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On)
15	7, 14	jaoi 853	. 2 ⊢ ((𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏) → (𝐻‘𝑎) ∈ On)
16	1, 15	syl 17	1 ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On)

Syntax hints:   → wi 4   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  Vcvv 3486  ∅c0 4279  𝒫 cpw 4525   ↦ cmpt 5132   × cxp 5539   ↾ cres 5543  Oncon0 6177  suc csuc 6179  ‘cfv 6341  ωcom 7566  reccrdg 8031  harchar 9006

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-pss 3942  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-tp 4558  df-op 4560  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5446  df-eprel 5451  df-po 5460  df-so 5461  df-fr 5500  df-se 5501  df-we 5502  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-pred 6134  df-ord 6180  df-on 6181  df-lim 6182  df-suc 6183  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-isom 6350  df-riota 7100  df-om 7567  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-en 8496  df-dom 8497  df-oi 8960  df-har 9008

This theorem is referenced by:  hsmexlem4  9837  hsmexlem5  9838