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Mirrors > Home > MPE Home > Th. List > hsmexlem9 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 10433. 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 7890 | . 2 ⊢ (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏)) | |
2 | fveq2 6891 | . . . 4 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) = (𝐻‘∅)) | |
3 | hsmexlem7.h | . . . . . 6 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) | |
4 | 3 | hsmexlem7 10424 | . . . . 5 ⊢ (𝐻‘∅) = (har‘𝒫 𝑋) |
5 | harcl 9560 | . . . . 5 ⊢ (har‘𝒫 𝑋) ∈ On | |
6 | 4, 5 | eqeltri 2828 | . . . 4 ⊢ (𝐻‘∅) ∈ On |
7 | 2, 6 | eqeltrdi 2840 | . . 3 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) ∈ On) |
8 | 3 | hsmexlem8 10425 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻‘𝑏)))) |
9 | harcl 9560 | . . . . . 6 ⊢ (har‘𝒫 (𝑋 × (𝐻‘𝑏))) ∈ On | |
10 | 8, 9 | eqeltrdi 2840 | . . . . 5 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On) |
11 | fveq2 6891 | . . . . . 6 ⊢ (𝑎 = suc 𝑏 → (𝐻‘𝑎) = (𝐻‘suc 𝑏)) | |
12 | 11 | eleq1d 2817 | . . . . 5 ⊢ (𝑎 = suc 𝑏 → ((𝐻‘𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On)) |
13 | 10, 12 | syl5ibrcom 246 | . . . 4 ⊢ (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On)) |
14 | 13 | rexlimiv 3147 | . . 3 ⊢ (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On) |
15 | 7, 14 | jaoi 854 | . 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 844 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 Vcvv 3473 ∅c0 4322 𝒫 cpw 4602 ↦ cmpt 5231 × cxp 5674 ↾ cres 5678 Oncon0 6364 suc csuc 6366 ‘cfv 6543 ωcom 7859 reccrdg 8415 harchar 9557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-en 8946 df-dom 8947 df-oi 9511 df-har 9558 |
This theorem is referenced by: hsmexlem4 10430 hsmexlem5 10431 |
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