![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ssltright | Structured version Visualization version GIF version |
Description: A surreal is less than its right options. Theorem 0(i) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.) |
Ref | Expression |
---|---|
ssltright | ⊢ (𝐴 ∈ No → {𝐴} <<s ( R ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5454 | . . 3 ⊢ {𝐴} ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ∈ V) |
3 | fvexd 6934 | . 2 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ∈ V) | |
4 | snssi 4833 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No ) | |
5 | rightf 27914 | . . . 4 ⊢ R : No ⟶𝒫 No | |
6 | 5 | ffvelcdmi 7115 | . . 3 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ∈ 𝒫 No ) |
7 | 6 | elpwid 4631 | . 2 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ No ) |
8 | velsn 4664 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
9 | rightval 27912 | . . . . . . . . . 10 ⊢ ( R ‘𝐴) = {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑦} | |
10 | 9 | a1i 11 | . . . . . . . . 9 ⊢ (𝐴 ∈ No → ( R ‘𝐴) = {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑦}) |
11 | 10 | eleq2d 2824 | . . . . . . . 8 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( R ‘𝐴) ↔ 𝑦 ∈ {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑦})) |
12 | rabid 3459 | . . . . . . . 8 ⊢ (𝑦 ∈ {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑦} ↔ (𝑦 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝑦)) | |
13 | 11, 12 | bitrdi 287 | . . . . . . 7 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( R ‘𝐴) ↔ (𝑦 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝑦))) |
14 | 13 | simplbda 499 | . . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑦) |
15 | breq1 5172 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦)) | |
16 | 14, 15 | imbitrrid 246 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝑥 <s 𝑦)) |
17 | 16 | expd 415 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ No → (𝑦 ∈ ( R ‘𝐴) → 𝑥 <s 𝑦))) |
18 | 8, 17 | sylbi 217 | . . 3 ⊢ (𝑥 ∈ {𝐴} → (𝐴 ∈ No → (𝑦 ∈ ( R ‘𝐴) → 𝑥 <s 𝑦))) |
19 | 18 | 3imp21 1114 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ {𝐴} ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝑥 <s 𝑦) |
20 | 2, 3, 4, 7, 19 | ssltd 27845 | 1 ⊢ (𝐴 ∈ No → {𝐴} <<s ( R ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 {crab 3438 Vcvv 3482 𝒫 cpw 4622 {csn 4648 class class class wbr 5169 ‘cfv 6572 No csur 27693 <s cslt 27694 bday cbday 27695 <<s csslt 27834 O cold 27891 R cright 27894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-1o 8518 df-2o 8519 df-no 27696 df-slt 27697 df-bday 27698 df-sslt 27835 df-scut 27837 df-made 27895 df-old 27896 df-right 27899 |
This theorem is referenced by: lltropt 27920 madebdaylemlrcut 27946 mulsproplem5 28155 mulsproplem6 28156 mulsproplem7 28157 mulsproplem8 28158 mulsuniflem 28184 |
Copyright terms: Public domain | W3C validator |