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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 5427 | . . 3 ⊢ {𝐴} ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ∈ V) |
3 | fvexd 6906 | . 2 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ∈ V) | |
4 | snssi 4807 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No ) | |
5 | rightf 27786 | . . . 4 ⊢ R : No ⟶𝒫 No | |
6 | 5 | ffvelcdmi 7087 | . . 3 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ∈ 𝒫 No ) |
7 | 6 | elpwid 4607 | . 2 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ No ) |
8 | velsn 4640 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
9 | rightval 27784 | . . . . . . . . . 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 2815 | . . . . . . . 8 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( R ‘𝐴) ↔ 𝑦 ∈ {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑦})) |
12 | rabid 3448 | . . . . . . . 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 5145 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦)) | |
16 | 14, 15 | imbitrrid 245 | . . . . 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 216 | . . 3 ⊢ (𝑥 ∈ {𝐴} → (𝐴 ∈ No → (𝑦 ∈ ( R ‘𝐴) → 𝑥 <s 𝑦))) |
19 | 18 | 3imp21 1112 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ {𝐴} ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝑥 <s 𝑦) |
20 | 2, 3, 4, 7, 19 | ssltd 27717 | 1 ⊢ (𝐴 ∈ No → {𝐴} <<s ( R ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3428 Vcvv 3470 𝒫 cpw 4598 {csn 4624 class class class wbr 5142 ‘cfv 6542 No csur 27566 <s cslt 27567 bday cbday 27568 <<s csslt 27706 O cold 27763 R cright 27766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-1o 8480 df-2o 8481 df-no 27569 df-slt 27570 df-bday 27571 df-sslt 27707 df-scut 27709 df-made 27767 df-old 27768 df-right 27771 |
This theorem is referenced by: lltropt 27792 madebdaylemlrcut 27818 mulsproplem5 28013 mulsproplem6 28014 mulsproplem7 28015 mulsproplem8 28016 mulsuniflem 28042 |
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