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| 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 27690 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
| 2 | frn 6743 | . . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o}) | |
| 3 | 2 | rexlimivw 3151 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o}) | 
| 4 | 1, 3 | sylbi 217 | 1 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 {cpr 4628 ran crn 5686 Oncon0 6384 ⟶wf 6557 1oc1o 8499 2oc2o 8500 No csur 27684 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-no 27687 | 
| This theorem is referenced by: elno2 27699 nofv 27702 sltres 27707 noextend 27711 noextendseq 27712 nosepssdm 27731 nodenselem8 27736 nolt02olem 27739 nosupno 27748 noinfno 27763 | 
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