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| Mirrors > Home > MPE Home > Th. List > 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 27595 | . 2 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On) | |
| 2 | eloni 6330 | . 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 2109 dom cdm 5631 Ord word 6319 Oncon0 6320 No csur 27584 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-tr 5210 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-ord 6323 df-on 6324 df-fun 6501 df-fn 6502 df-f 6503 df-no 27587 |
| This theorem is referenced by: noseponlem 27609 nosepon 27610 noextend 27611 noextenddif 27613 noextendlt 27614 noextendgt 27615 nolesgn2ores 27617 nogesgn1ores 27619 fvnobday 27623 nosepssdm 27631 nosupres 27652 nosupbnd1lem1 27653 nosupbnd1lem3 27655 nosupbnd1lem5 27657 nosupbnd2lem1 27660 nosupbnd2 27661 noinfres 27667 noinfbnd1lem1 27668 noinfbnd1lem3 27670 noinfbnd1lem5 27672 noinfbnd2lem1 27675 noinfbnd2 27676 noetasuplem4 27681 noetainflem4 27685 noetalem1 27686 |
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