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Mirrors > Home > MPE Home > Th. List > hsmexlem9 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 10011. 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 7651 | . 2 ⊢ (𝑎 ∈ ω → (𝑎 = ∅ ∨ ∃𝑏 ∈ ω 𝑎 = suc 𝑏)) | |
2 | fveq2 6695 | . . . 4 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) = (𝐻‘∅)) | |
3 | hsmexlem7.h | . . . . . 6 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) | |
4 | 3 | hsmexlem7 10002 | . . . . 5 ⊢ (𝐻‘∅) = (har‘𝒫 𝑋) |
5 | harcl 9153 | . . . . 5 ⊢ (har‘𝒫 𝑋) ∈ On | |
6 | 4, 5 | eqeltri 2827 | . . . 4 ⊢ (𝐻‘∅) ∈ On |
7 | 2, 6 | eqeltrdi 2839 | . . 3 ⊢ (𝑎 = ∅ → (𝐻‘𝑎) ∈ On) |
8 | 3 | hsmexlem8 10003 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) = (har‘𝒫 (𝑋 × (𝐻‘𝑏)))) |
9 | harcl 9153 | . . . . . 6 ⊢ (har‘𝒫 (𝑋 × (𝐻‘𝑏))) ∈ On | |
10 | 8, 9 | eqeltrdi 2839 | . . . . 5 ⊢ (𝑏 ∈ ω → (𝐻‘suc 𝑏) ∈ On) |
11 | fveq2 6695 | . . . . . 6 ⊢ (𝑎 = suc 𝑏 → (𝐻‘𝑎) = (𝐻‘suc 𝑏)) | |
12 | 11 | eleq1d 2815 | . . . . 5 ⊢ (𝑎 = suc 𝑏 → ((𝐻‘𝑎) ∈ On ↔ (𝐻‘suc 𝑏) ∈ On)) |
13 | 10, 12 | syl5ibrcom 250 | . . . 4 ⊢ (𝑏 ∈ ω → (𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On)) |
14 | 13 | rexlimiv 3189 | . . 3 ⊢ (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → (𝐻‘𝑎) ∈ On) |
15 | 7, 14 | jaoi 857 | . 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 847 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 Vcvv 3398 ∅c0 4223 𝒫 cpw 4499 ↦ cmpt 5120 × cxp 5534 ↾ cres 5538 Oncon0 6191 suc csuc 6193 ‘cfv 6358 ωcom 7622 reccrdg 8123 harchar 9150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-en 8605 df-dom 8606 df-oi 9104 df-har 9151 |
This theorem is referenced by: hsmexlem4 10008 hsmexlem5 10009 |
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