MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem9 Structured version   Visualization version   GIF version

Theorem hsmexlem9 10465
Description: Lemma for hsmex 10472. 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 7916 . 2 (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏))
2 fveq2 6906 . . . 4 (𝑎 = ∅ → (𝐻𝑎) = (𝐻‘∅))
3 hsmexlem7.h . . . . . 6 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
43hsmexlem7 10463 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
5 harcl 9599 . . . . 5 (har‘𝒫 𝑋) ∈ On
64, 5eqeltri 2837 . . . 4 (𝐻‘∅) ∈ On
72, 6eqeltrdi 2849 . . 3 (𝑎 = ∅ → (𝐻𝑎) ∈ On)
83hsmexlem8 10464 . . . . . 6 (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻𝑏))))
9 harcl 9599 . . . . . 6 (har‘𝒫 (𝑋 × (𝐻𝑏))) ∈ On
108, 9eqeltrdi 2849 . . . . 5 (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On)
11 fveq2 6906 . . . . . 6 (𝑎 = suc 𝑏 → (𝐻𝑎) = (𝐻‘suc 𝑏))
1211eleq1d 2826 . . . . 5 (𝑎 = suc 𝑏 → ((𝐻𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On))
1310, 12syl5ibrcom 247 . . . 4 (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On))
1413rexlimiv 3148 . . 3 (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻𝑎) ∈ On)
157, 14jaoi 858 . 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 848   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  c0 4333  𝒫 cpw 4600  cmpt 5225   × cxp 5683  cres 5687  Oncon0 6384  suc csuc 6386  cfv 6561  ωcom 7887  reccrdg 8449  harchar 9596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-en 8986  df-dom 8987  df-oi 9550  df-har 9597
This theorem is referenced by:  hsmexlem4  10469  hsmexlem5  10470
  Copyright terms: Public domain W3C validator