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Mirrors > Home > MPE Home > Th. List > om2uzoi | Structured version Visualization version GIF version |
Description: An alternative definition of 𝐺 in terms of df-oi 8973. (Contributed by Mario Carneiro, 2-Jun-2015.) |
Ref | Expression |
---|---|
om2uz.1 | ⊢ 𝐶 ∈ ℤ |
om2uz.2 | ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) |
Ref | Expression |
---|---|
om2uzoi | ⊢ 𝐺 = OrdIso( < , (ℤ≥‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7588 | . . . 4 ⊢ Ord ω | |
2 | om2uz.1 | . . . . 5 ⊢ 𝐶 ∈ ℤ | |
3 | om2uz.2 | . . . . 5 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
4 | 2, 3 | om2uzisoi 13321 | . . . 4 ⊢ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶)) |
5 | 1, 4 | pm3.2i 473 | . . 3 ⊢ (Ord ω ∧ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶))) |
6 | ordwe 6203 | . . . . . 6 ⊢ (Ord ω → E We ω) | |
7 | 1, 6 | ax-mp 5 | . . . . 5 ⊢ E We ω |
8 | isowe 7101 | . . . . . 6 ⊢ (𝐺 Isom E , < (ω, (ℤ≥‘𝐶)) → ( E We ω ↔ < We (ℤ≥‘𝐶))) | |
9 | 4, 8 | ax-mp 5 | . . . . 5 ⊢ ( E We ω ↔ < We (ℤ≥‘𝐶)) |
10 | 7, 9 | mpbi 232 | . . . 4 ⊢ < We (ℤ≥‘𝐶) |
11 | fvex 6682 | . . . . 5 ⊢ (ℤ≥‘𝐶) ∈ V | |
12 | exse 5518 | . . . . 5 ⊢ ((ℤ≥‘𝐶) ∈ V → < Se (ℤ≥‘𝐶)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ < Se (ℤ≥‘𝐶) |
14 | eqid 2821 | . . . . 5 ⊢ OrdIso( < , (ℤ≥‘𝐶)) = OrdIso( < , (ℤ≥‘𝐶)) | |
15 | 14 | oieu 9002 | . . . 4 ⊢ (( < We (ℤ≥‘𝐶) ∧ < Se (ℤ≥‘𝐶)) → ((Ord ω ∧ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶))) ↔ (ω = dom OrdIso( < , (ℤ≥‘𝐶)) ∧ 𝐺 = OrdIso( < , (ℤ≥‘𝐶))))) |
16 | 10, 13, 15 | mp2an 690 | . . 3 ⊢ ((Ord ω ∧ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶))) ↔ (ω = dom OrdIso( < , (ℤ≥‘𝐶)) ∧ 𝐺 = OrdIso( < , (ℤ≥‘𝐶)))) |
17 | 5, 16 | mpbi 232 | . 2 ⊢ (ω = dom OrdIso( < , (ℤ≥‘𝐶)) ∧ 𝐺 = OrdIso( < , (ℤ≥‘𝐶))) |
18 | 17 | simpri 488 | 1 ⊢ 𝐺 = OrdIso( < , (ℤ≥‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ↦ cmpt 5145 E cep 5463 Se wse 5511 We wwe 5512 dom cdm 5554 ↾ cres 5556 Ord word 6189 ‘cfv 6354 Isom wiso 6355 (class class class)co 7155 ωcom 7579 reccrdg 8044 OrdIsocoi 8972 1c1 10537 + caddc 10539 < clt 10674 ℤcz 11980 ℤ≥cuz 12242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-oi 8973 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 |
This theorem is referenced by: ltbwe 20252 |
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