Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 33159 | . 2 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On) | |
2 | eloni 6203 | . 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 2114 dom cdm 5557 Ord word 6192 Oncon0 6193 No csur 33149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-no 33152 |
This theorem is referenced by: noseponlem 33173 nosepon 33174 noextend 33175 noextenddif 33177 noextendlt 33178 noextendgt 33179 nolesgn2ores 33181 fvnobday 33185 nosepssdm 33192 nosupbday 33207 nosupres 33209 nosupbnd1lem1 33210 nosupbnd1lem3 33212 nosupbnd1lem5 33214 nosupbnd2lem1 33217 nosupbnd2 33218 noetalem3 33221 |
Copyright terms: Public domain | W3C validator |