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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 27149 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
2 | fdm 6727 | . . . . 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 4631 dom cdm 5677 Oncon0 6365 ⟶wf 6540 1oc1o 8459 2oc2o 8460 No csur 27143 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-no 27146 |
This theorem is referenced by: nodmord 27156 elno2 27157 noseponlem 27167 noextend 27169 noextendseq 27170 noextenddif 27171 noextendlt 27172 noextendgt 27173 bdayfo 27180 nosepssdm 27189 nolt02olem 27197 nosupno 27206 nosupres 27210 nosupbnd1lem1 27211 nosupbnd1lem2 27212 nosupbnd1lem3 27213 nosupbnd1lem4 27214 nosupbnd1lem5 27215 nosupbnd1lem6 27216 nosupbnd1 27217 nosupbnd2lem1 27218 nosupbnd2 27219 noinfno 27221 noinfres 27225 noinfbnd1lem1 27226 noinfbnd1lem2 27227 noinfbnd1lem3 27228 noinfbnd1lem4 27229 noinfbnd1lem5 27230 noinfbnd1lem6 27231 noinfbnd1 27232 noinfbnd2lem1 27233 noinfbnd2 27234 nosupinfsep 27235 noetasuplem3 27238 noetasuplem4 27239 noetainflem3 27242 noetainflem4 27243 bdaybndex 42182 |
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