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Mirrors > Home > MPE Home > Th. List > nodmon | Structured version Visualization version GIF version |
Description: The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.) |
Ref | Expression |
---|---|
nodmon | ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elno 27139 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
2 | fdm 6724 | . . . . 5 ⊢ (𝐴:𝑥⟶{1o, 2o} → dom 𝐴 = 𝑥) | |
3 | 2 | eleq1d 2819 | . . . 4 ⊢ (𝐴:𝑥⟶{1o, 2o} → (dom 𝐴 ∈ On ↔ 𝑥 ∈ On)) |
4 | 3 | biimprcd 249 | . . 3 ⊢ (𝑥 ∈ On → (𝐴:𝑥⟶{1o, 2o} → dom 𝐴 ∈ On)) |
5 | 4 | rexlimiv 3149 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → dom 𝐴 ∈ On) |
6 | 1, 5 | sylbi 216 | 1 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∃wrex 3071 {cpr 4630 dom cdm 5676 Oncon0 6362 ⟶wf 6537 1oc1o 8456 2oc2o 8457 No csur 27133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-no 27136 |
This theorem is referenced by: nodmord 27146 elno2 27147 noseponlem 27157 noextend 27159 noextendseq 27160 noextenddif 27161 noextendlt 27162 noextendgt 27163 bdayfo 27170 nosepssdm 27179 nolt02olem 27187 nosupno 27196 nosupres 27200 nosupbnd1lem1 27201 nosupbnd1lem2 27202 nosupbnd1lem3 27203 nosupbnd1lem4 27204 nosupbnd1lem5 27205 nosupbnd1lem6 27206 nosupbnd1 27207 nosupbnd2lem1 27208 nosupbnd2 27209 noinfno 27211 noinfres 27215 noinfbnd1lem1 27216 noinfbnd1lem2 27217 noinfbnd1lem3 27218 noinfbnd1lem4 27219 noinfbnd1lem5 27220 noinfbnd1lem6 27221 noinfbnd1 27222 noinfbnd2lem1 27223 noinfbnd2 27224 nosupinfsep 27225 noetasuplem3 27228 noetasuplem4 27229 noetainflem3 27232 noetainflem4 27233 bdaybndex 42168 |
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