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Mirrors > Home > MPE Home > Th. List > Mathboxes > negsf1o | Structured version Visualization version GIF version |
Description: Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.) |
Ref | Expression |
---|---|
negsf1o | ⊢ -us : No –1-1-onto→ No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negsf 34333 | . . 3 ⊢ -us : No ⟶ No | |
2 | negs11 34332 | . . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (( -us ‘𝑥) = ( -us ‘𝑦) ↔ 𝑥 = 𝑦)) | |
3 | 2 | biimpd 228 | . . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (( -us ‘𝑥) = ( -us ‘𝑦) → 𝑥 = 𝑦)) |
4 | 3 | rgen2 3192 | . . 3 ⊢ ∀𝑥 ∈ No ∀𝑦 ∈ No (( -us ‘𝑥) = ( -us ‘𝑦) → 𝑥 = 𝑦) |
5 | dff13 7198 | . . 3 ⊢ ( -us : No –1-1→ No ↔ ( -us : No ⟶ No ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (( -us ‘𝑥) = ( -us ‘𝑦) → 𝑥 = 𝑦))) | |
6 | 1, 4, 5 | mpbir2an 709 | . 2 ⊢ -us : No –1-1→ No |
7 | negsfo 34334 | . 2 ⊢ -us : No –onto→ No | |
8 | df-f1o 6500 | . 2 ⊢ ( -us : No –1-1-onto→ No ↔ ( -us : No –1-1→ No ∧ -us : No –onto→ No )) | |
9 | 6, 7, 8 | mpbir2an 709 | 1 ⊢ -us : No –1-1-onto→ No |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ⟶wf 6489 –1-1→wf1 6490 –onto→wfo 6491 –1-1-onto→wf1o 6492 ‘cfv 6493 No csur 26939 -us cnegs 34306 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-1o 8404 df-2o 8405 df-no 26942 df-slt 26943 df-bday 26944 df-sle 27044 df-sslt 27072 df-scut 27074 df-0s 27114 df-made 27128 df-old 27129 df-left 27131 df-right 27132 df-nadd 34215 df-norec 34246 df-norec2 34257 df-adds 34268 df-negs 34308 |
This theorem is referenced by: (None) |
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