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Theorem hsmexlem9 9130
 Description: Lemma for hsmex 9137. 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 6982 . 2 (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏))
2 fveq2 6103 . . . 4 (𝑎 = ∅ → (𝐻𝑎) = (𝐻‘∅))
3 hsmexlem7.h . . . . . 6 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
43hsmexlem7 9128 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
5 harcl 8349 . . . . 5 (har‘𝒫 𝑋) ∈ On
64, 5eqeltri 2684 . . . 4 (𝐻‘∅) ∈ On
72, 6syl6eqel 2696 . . 3 (𝑎 = ∅ → (𝐻𝑎) ∈ On)
83hsmexlem8 9129 . . . . . 6 (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻𝑏))))
9 harcl 8349 . . . . . 6 (har‘𝒫 (𝑋 × (𝐻𝑏))) ∈ On
108, 9syl6eqel 2696 . . . . 5 (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On)
11 fveq2 6103 . . . . . 6 (𝑎 = suc 𝑏 → (𝐻𝑎) = (𝐻‘suc 𝑏))
1211eleq1d 2672 . . . . 5 (𝑎 = suc 𝑏 → ((𝐻𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On))
1310, 12syl5ibrcom 236 . . . 4 (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On))
1413rexlimiv 3009 . . 3 (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On)
157, 14jaoi 393 . 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 382   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040  Oncon0 5640  suc csuc 5642  ‘cfv 5804  ωcom 6957  reccrdg 7392  harchar 8344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-en 7842  df-dom 7843  df-oi 8298  df-har 8346 This theorem is referenced by:  hsmexlem4  9134  hsmexlem5  9135
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