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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 27498 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
2 | frn 6715 | . . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o}) | |
3 | 2 | rexlimivw 3143 | . 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 2098 ∃wrex 3062 ⊆ wss 3941 {cpr 4623 ran crn 5668 Oncon0 6355 ⟶wf 6530 1oc1o 8455 2oc2o 8456 No csur 27492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-no 27495 |
This theorem is referenced by: elno2 27506 nofv 27509 sltres 27514 noextend 27518 noextendseq 27519 nosepssdm 27538 nodenselem8 27543 nolt02olem 27546 nosupno 27555 noinfno 27570 |
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