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Mirrors > Home > MPE Home > Th. List > Mathboxes > nodmord | Structured version Visualization version GIF version |
Description: The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.) |
Ref | Expression |
---|---|
nodmord | ⊢ (𝐴 ∈ No → Ord dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nodmon 33157 | . 2 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On) | |
2 | eloni 6200 | . 2 ⊢ (dom 𝐴 ∈ On → Ord dom 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ No → Ord dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 dom cdm 5554 Ord word 6189 Oncon0 6190 No csur 33147 |
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-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 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-po 5473 df-so 5474 df-fr 5513 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-ord 6193 df-on 6194 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-no 33150 |
This theorem is referenced by: noseponlem 33171 nosepon 33172 noextend 33173 noextenddif 33175 noextendlt 33176 noextendgt 33177 nolesgn2ores 33179 fvnobday 33183 nosepssdm 33190 nosupbday 33205 nosupres 33207 nosupbnd1lem1 33208 nosupbnd1lem3 33210 nosupbnd1lem5 33212 nosupbnd2lem1 33215 nosupbnd2 33216 noetalem3 33219 |
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