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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 8987 | . 2 ⊢ Ord dom 𝐹 |
3 | 1 | oiexg 8993 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
4 | dmexg 7607 | . . 3 ⊢ (𝐹 ∈ V → dom 𝐹 ∈ V) | |
5 | elong 6194 | . . 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 260 | 1 ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3495 dom cdm 5550 Ord word 6185 Oncon0 6186 OrdIsocoi 8967 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
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-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-wrecs 7941 df-recs 8002 df-oi 8968 |
This theorem is referenced by: hartogslem1 9000 wofib 9003 cantnfcl 9124 cantnflt2 9130 cantnflem1 9146 wemapwe 9154 cnfcom2 9159 cnfcom3lem 9160 cnfcom3 9161 finnisoeu 9533 dfac12lem2 9564 cofsmo 9685 pwfseqlem5 10079 fz1isolem 13813 |
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