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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 8977 | . 2 ⊢ Ord dom 𝐹 |
3 | 1 | oiexg 8983 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
4 | dmexg 7594 | . . 3 ⊢ (𝐹 ∈ V → dom 𝐹 ∈ V) | |
5 | elong 6167 | . . 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 261 | 1 ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 dom cdm 5519 Ord word 6158 Oncon0 6159 OrdIsocoi 8957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-wrecs 7930 df-recs 7991 df-oi 8958 |
This theorem is referenced by: hartogslem1 8990 wofib 8993 cantnfcl 9114 cantnflt2 9120 cantnflem1 9136 wemapwe 9144 cnfcom2 9149 cnfcom3lem 9150 cnfcom3 9151 finnisoeu 9524 dfac12lem2 9555 cofsmo 9680 pwfseqlem5 10074 fz1isolem 13815 |
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