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| Mirrors > Home > MPE Home > Th. List > om2noseqoi | Structured version Visualization version GIF version | ||
| Description: An alternative definition of 𝐺 in terms of df-oi 9460. (Contributed by Scott Fenton, 18-Apr-2025.) |
| Ref | Expression |
|---|---|
| om2noseq.1 | ⊢ (𝜑 → 𝐶 ∈ No ) |
| om2noseq.2 | ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)) |
| om2noseq.3 | ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)) |
| Ref | Expression |
|---|---|
| om2noseqoi | ⊢ (𝜑 → 𝐺 = OrdIso( <s , 𝑍)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | om2noseq.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ No ) | |
| 2 | om2noseq.2 | . . . . 5 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)) | |
| 3 | om2noseq.3 | . . . . 5 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)) | |
| 4 | 1, 2, 3 | om2noseqiso 28449 | . . . 4 ⊢ (𝜑 → 𝐺 Isom E , <s (ω, 𝑍)) |
| 5 | ordom 7860 | . . . 4 ⊢ Ord ω | |
| 6 | 4, 5 | jctil 528 | . . 3 ⊢ (𝜑 → (Ord ω ∧ 𝐺 Isom E , <s (ω, 𝑍))) |
| 7 | ordwe 6362 | . . . . . 6 ⊢ (Ord ω → E We ω) | |
| 8 | 5, 7 | ax-mp 5 | . . . . 5 ⊢ E We ω |
| 9 | isowe 7337 | . . . . . 6 ⊢ (𝐺 Isom E , <s (ω, 𝑍) → ( E We ω ↔ <s We 𝑍)) | |
| 10 | 4, 9 | syl 18 | . . . . 5 ⊢ (𝜑 → ( E We ω ↔ <s We 𝑍)) |
| 11 | 8, 10 | mpbii 236 | . . . 4 ⊢ (𝜑 → <s We 𝑍) |
| 12 | 3 | noseqex 28436 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ V) |
| 13 | exse 5611 | . . . . 5 ⊢ (𝑍 ∈ V → <s Se 𝑍) | |
| 14 | 12, 13 | syl 18 | . . . 4 ⊢ (𝜑 → <s Se 𝑍) |
| 15 | eqid 2765 | . . . . 5 ⊢ OrdIso( <s , 𝑍) = OrdIso( <s , 𝑍) | |
| 16 | 15 | oieu 9489 | . . . 4 ⊢ (( <s We 𝑍 ∧ <s Se 𝑍) → ((Ord ω ∧ 𝐺 Isom E , <s (ω, 𝑍)) ↔ (ω = dom OrdIso( <s , 𝑍) ∧ 𝐺 = OrdIso( <s , 𝑍)))) |
| 17 | 11, 14, 16 | syl2anc 595 | . . 3 ⊢ (𝜑 → ((Ord ω ∧ 𝐺 Isom E , <s (ω, 𝑍)) ↔ (ω = dom OrdIso( <s , 𝑍) ∧ 𝐺 = OrdIso( <s , 𝑍)))) |
| 18 | 6, 17 | mpbid 235 | . 2 ⊢ (𝜑 → (ω = dom OrdIso( <s , 𝑍) ∧ 𝐺 = OrdIso( <s , 𝑍))) |
| 19 | 18 | simprd 500 | 1 ⊢ (𝜑 → 𝐺 = OrdIso( <s , 𝑍)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ↦ cmpt 5185 E cep 5550 Se wse 5602 We wwe 5603 dom cdm 5651 ↾ cres 5653 “ cima 5654 Ord word 6348 Isom wiso 6526 (class class class)co 7400 ωcom 7850 reccrdg 8384 OrdIsocoi 9459 No csur 27758 <s clts 27759 1s c1s 27953 +s cadds 28106 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-dc 10418 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4908 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-nadd 8640 df-oi 9460 df-no 27761 df-lts 27762 df-bday 27763 df-les 27863 df-slts 27905 df-cuts 27907 df-0s 27954 df-1s 27955 df-made 27974 df-old 27975 df-left 27977 df-right 27978 df-norec2 28096 df-adds 28107 |
| This theorem is referenced by: (None) |
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