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Mirrors > Home > MPE Home > Th. List > oion | Structured version Visualization version GIF version |
Description: The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 23-May-2015.) |
Ref | Expression |
---|---|
oicl.1 | ⊢ 𝐹 = OrdIso(𝑅, 𝐴) |
Ref | Expression |
---|---|
oion | ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oicl.1 | . . 3 ⊢ 𝐹 = OrdIso(𝑅, 𝐴) | |
2 | 1 | oicl 8723 | . 2 ⊢ Ord dom 𝐹 |
3 | 1 | oiexg 8729 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
4 | dmexg 7375 | . . 3 ⊢ (𝐹 ∈ V → dom 𝐹 ∈ V) | |
5 | elong 5984 | . . 3 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∈ On ↔ Ord dom 𝐹)) | |
6 | 3, 4, 5 | 3syl 18 | . 2 ⊢ (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ On ↔ Ord dom 𝐹)) |
7 | 2, 6 | mpbiri 250 | 1 ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 Vcvv 3397 dom cdm 5355 Ord word 5975 Oncon0 5976 OrdIsocoi 8703 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-wrecs 7689 df-recs 7751 df-oi 8704 |
This theorem is referenced by: hartogslem1 8736 wofib 8739 cantnfcl 8861 cantnflt2 8867 cantnflem1 8883 wemapwe 8891 cnfcom2 8896 cnfcom3lem 8897 cnfcom3 8898 finnisoeu 9269 dfac12lem2 9301 cofsmo 9426 pwfseqlem5 9820 fz1isolem 13559 |
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