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Mirrors > Home > MPE Home > Th. List > ssltleft | Structured version Visualization version GIF version |
Description: A surreal is greater than its left options. Theorem 0(ii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.) |
Ref | Expression |
---|---|
ssltleft | ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6911 | . 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ∈ V) | |
2 | snex 5433 | . . 3 ⊢ {𝐴} ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ∈ V) |
4 | leftf 27843 | . . . 4 ⊢ L : No ⟶𝒫 No | |
5 | 4 | ffvelcdmi 7092 | . . 3 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ∈ 𝒫 No ) |
6 | 5 | elpwid 4613 | . 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No ) |
7 | snssi 4813 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No ) | |
8 | velsn 4646 | . . . 4 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
9 | leftval 27841 | . . . . . . . . . 10 ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} | |
10 | 9 | a1i 11 | . . . . . . . . 9 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}) |
11 | 10 | eleq2d 2811 | . . . . . . . 8 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})) |
12 | rabid 3439 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ↔ (𝑥 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝑥 <s 𝐴)) | |
13 | 11, 12 | bitrdi 286 | . . . . . . 7 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) ↔ (𝑥 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝑥 <s 𝐴))) |
14 | 13 | simplbda 498 | . . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘𝐴)) → 𝑥 <s 𝐴) |
15 | breq2 5153 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴)) | |
16 | 14, 15 | imbitrrid 245 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘𝐴)) → 𝑥 <s 𝑦)) |
17 | 16 | expd 414 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) → 𝑥 <s 𝑦))) |
18 | 8, 17 | sylbi 216 | . . 3 ⊢ (𝑦 ∈ {𝐴} → (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) → 𝑥 <s 𝑦))) |
19 | 18 | 3imp231 1110 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘𝐴) ∧ 𝑦 ∈ {𝐴}) → 𝑥 <s 𝑦) |
20 | 1, 3, 6, 7, 19 | ssltd 27775 | 1 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3418 Vcvv 3461 𝒫 cpw 4604 {csn 4630 class class class wbr 5149 ‘cfv 6549 No csur 27623 <s cslt 27624 bday cbday 27625 <<s csslt 27764 O cold 27821 L cleft 27823 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-1o 8487 df-2o 8488 df-no 27626 df-slt 27627 df-bday 27628 df-sslt 27765 df-scut 27767 df-made 27825 df-old 27826 df-left 27828 |
This theorem is referenced by: lltropt 27850 madebdaylemlrcut 27876 mulsproplem5 28075 mulsproplem6 28076 mulsproplem7 28077 mulsproplem8 28078 mulsuniflem 28104 |
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