![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hsmexlem9 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 9591. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Ref | Expression |
---|---|
hsmexlem7.h | ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) |
Ref | Expression |
---|---|
hsmexlem9 | ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0suc 7370 | . 2 ⊢ (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏)) | |
2 | fveq2 6448 | . . . 4 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) = (𝐻‘∅)) | |
3 | hsmexlem7.h | . . . . . 6 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) | |
4 | 3 | hsmexlem7 9582 | . . . . 5 ⊢ (𝐻‘∅) = (har‘𝒫 𝑋) |
5 | harcl 8757 | . . . . 5 ⊢ (har‘𝒫 𝑋) ∈ On | |
6 | 4, 5 | eqeltri 2855 | . . . 4 ⊢ (𝐻‘∅) ∈ On |
7 | 2, 6 | syl6eqel 2867 | . . 3 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) ∈ On) |
8 | 3 | hsmexlem8 9583 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻‘𝑏)))) |
9 | harcl 8757 | . . . . . 6 ⊢ (har‘𝒫 (𝑋 × (𝐻‘𝑏))) ∈ On | |
10 | 8, 9 | syl6eqel 2867 | . . . . 5 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On) |
11 | fveq2 6448 | . . . . . 6 ⊢ (𝑎 = suc 𝑏 → (𝐻‘𝑎) = (𝐻‘suc 𝑏)) | |
12 | 11 | eleq1d 2844 | . . . . 5 ⊢ (𝑎 = suc 𝑏 → ((𝐻‘𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On)) |
13 | 10, 12 | syl5ibrcom 239 | . . . 4 ⊢ (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On)) |
14 | 13 | rexlimiv 3209 | . . 3 ⊢ (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On) |
15 | 7, 14 | jaoi 846 | . 2 ⊢ ((𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏) → (𝐻‘𝑎) ∈ On) |
16 | 1, 15 | syl 17 | 1 ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 Vcvv 3398 ∅c0 4141 𝒫 cpw 4379 ↦ cmpt 4967 × cxp 5355 ↾ cres 5359 Oncon0 5978 suc csuc 5980 ‘cfv 6137 ωcom 7345 reccrdg 7790 harchar 8752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-en 8244 df-dom 8245 df-oi 8706 df-har 8754 |
This theorem is referenced by: hsmexlem4 9588 hsmexlem5 9589 |
Copyright terms: Public domain | W3C validator |