Theorem hsmexlem9 10409

Description: Lemma for hsmex 10416. 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 7891	. 2 ⊢ (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏))
2		fveq2 6882	. . . 4 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) = (𝐻‘∅))
3		hsmexlem7.h	. . . . . 6 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
4	3	hsmexlem7 10407	. . . . 5 ⊢ (𝐻‘∅) = (har‘𝒫 𝑋)
5		harcl 9521	. . . . 5 ⊢ (har‘𝒫 𝑋) ∈ On
6	4, 5	eqeltri 2865	. . . 4 ⊢ (𝐻‘∅) ∈ On
7	2, 6	eqeltrdi 2877	. . 3 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) ∈ On)
8	3	hsmexlem8 10408	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻‘𝑏))))
9		harcl 9521	. . . . . 6 ⊢ (har‘𝒫 (𝑋 × (𝐻‘𝑏))) ∈ On
10	8, 9	eqeltrdi 2877	. . . . 5 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On)
11		fveq2 6882	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → (𝐻‘𝑎) = (𝐻‘suc 𝑏))
12	11	eleq1d 2854	. . . . 5 ⊢ (𝑎 = suc 𝑏 → ((𝐻‘𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On))
13	10, 12	syl5ibrcom 250	. . . 4 ⊢ (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On))
14	13	rexlimiv 3165	. . 3 ⊢ (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On)
15	7, 14	jaoi 870	. 2 ⊢ ((𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏) → (𝐻‘𝑎) ∈ On)
16	1, 15	syl 18	1 ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463  ∅c0 4294  𝒫 cpw 4567   ↦ cmpt 5196   × cxp 5660   ↾ cres 5664  Oncon0 6361  suc csuc 6363  ‘cfv 6537  ωcom 7862  reccrdg 8396  harchar 9518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-en 8944  df-dom 8945  df-oi 9472  df-har 9519

This theorem is referenced by:  hsmexlem4  10413  hsmexlem5  10414