Theorem hsmexlem9 9247

Description: Lemma for hsmex 9254. 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 7090	. 2 ⊢ (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏))
2		fveq2 6191	. . . 4 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) = (𝐻‘∅))
3		hsmexlem7.h	. . . . . 6 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
4	3	hsmexlem7 9245	. . . . 5 ⊢ (𝐻‘∅) = (har‘𝒫 𝑋)
5		harcl 8466	. . . . 5 ⊢ (har‘𝒫 𝑋) ∈ On
6	4, 5	eqeltri 2697	. . . 4 ⊢ (𝐻‘∅) ∈ On
7	2, 6	syl6eqel 2709	. . 3 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) ∈ On)
8	3	hsmexlem8 9246	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻‘𝑏))))
9		harcl 8466	. . . . . 6 ⊢ (har‘𝒫 (𝑋 × (𝐻‘𝑏))) ∈ On
10	8, 9	syl6eqel 2709	. . . . 5 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On)
11		fveq2 6191	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → (𝐻‘𝑎) = (𝐻‘suc 𝑏))
12	11	eleq1d 2686	. . . . 5 ⊢ (𝑎 = suc 𝑏 → ((𝐻‘𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On))
13	10, 12	syl5ibrcom 237	. . . 4 ⊢ (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On))
14	13	rexlimiv 3027	. . 3 ⊢ (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On)
15	7, 14	jaoi 394	. 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 383   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200  ∅c0 3915  𝒫 cpw 4158   ↦ cmpt 4729   × cxp 5112   ↾ cres 5116  Oncon0 5723  suc csuc 5725  ‘cfv 5888  ωcom 7065  reccrdg 7505  harchar 8461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-en 7956  df-dom 7957  df-oi 8415  df-har 8463

This theorem is referenced by:  hsmexlem4  9251  hsmexlem5  9252