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Mirrors > Home > MPE Home > Th. List > Mathboxes > norn | Structured version Visualization version GIF version |
Description: The range of a surreal is a subset of the surreal signs. (Contributed by Scott Fenton, 16-Jun-2011.) |
Ref | Expression |
---|---|
norn | ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1𝑜, 2𝑜}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elno 32101 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜}) | |
2 | frn 6210 | . . 3 ⊢ (𝐴:𝑥⟶{1𝑜, 2𝑜} → ran 𝐴 ⊆ {1𝑜, 2𝑜}) | |
3 | 2 | rexlimivw 3163 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜} → ran 𝐴 ⊆ {1𝑜, 2𝑜}) |
4 | 1, 3 | sylbi 207 | 1 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1𝑜, 2𝑜}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2135 ∃wrex 3047 ⊆ wss 3711 {cpr 4319 ran crn 5263 Oncon0 5880 ⟶wf 6041 1𝑜c1o 7718 2𝑜c2o 7719 No csur 32095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pr 5051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-ral 3051 df-rex 3052 df-reu 3053 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-nul 4055 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4585 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-id 5170 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-no 32098 |
This theorem is referenced by: elno2 32109 nofv 32112 sltres 32117 noextend 32121 noextendseq 32122 nosepssdm 32138 nodenselem8 32143 nolt02olem 32146 nosupno 32151 |
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