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Mirrors > Home > MPE Home > Th. List > 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 26992 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
2 | frn 6675 | . . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o}) | |
3 | 2 | rexlimivw 3148 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o}) |
4 | 1, 3 | sylbi 216 | 1 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∃wrex 3073 ⊆ wss 3910 {cpr 4588 ran crn 5634 Oncon0 6317 ⟶wf 6492 1oc1o 8404 2oc2o 8405 No csur 26986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-no 26989 |
This theorem is referenced by: elno2 27000 nofv 27003 sltres 27008 noextend 27012 noextendseq 27013 nosepssdm 27032 nodenselem8 27037 nolt02olem 27040 nosupno 27049 noinfno 27064 |
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