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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33050 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
2 | fdm 6515 | . . . . 5 ⊢ (𝐴:𝑥⟶{1o, 2o} → dom 𝐴 = 𝑥) | |
3 | 2 | eleq1d 2894 | . . . 4 ⊢ (𝐴:𝑥⟶{1o, 2o} → (dom 𝐴 ∈ On ↔ 𝑥 ∈ On)) |
4 | 3 | biimprcd 251 | . . 3 ⊢ (𝑥 ∈ On → (𝐴:𝑥⟶{1o, 2o} → dom 𝐴 ∈ On)) |
5 | 4 | rexlimiv 3277 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → dom 𝐴 ∈ On) |
6 | 1, 5 | sylbi 218 | 1 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∃wrex 3136 {cpr 4559 dom cdm 5548 Oncon0 6184 ⟶wf 6344 1oc1o 8084 2oc2o 8085 No csur 33044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-no 33047 |
This theorem is referenced by: nodmord 33057 elno2 33058 noseponlem 33068 noextend 33070 noextendseq 33071 noextenddif 33072 noextendlt 33073 noextendgt 33074 bdayfo 33079 nosepssdm 33087 nolt02olem 33095 nosupno 33100 nosupres 33104 nosupbnd1lem1 33105 nosupbnd1lem2 33106 nosupbnd1lem3 33107 nosupbnd1lem4 33108 nosupbnd1lem5 33109 nosupbnd1lem6 33110 nosupbnd1 33111 nosupbnd2lem1 33112 nosupbnd2 33113 noetalem2 33115 noetalem3 33116 |
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