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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 6921 | . 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ∈ V) | |
2 | snex 5441 | . . 3 ⊢ {𝐴} ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ∈ V) |
4 | leftf 27918 | . . . 4 ⊢ L : No ⟶𝒫 No | |
5 | 4 | ffvelcdmi 7102 | . . 3 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ∈ 𝒫 No ) |
6 | 5 | elpwid 4613 | . 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No ) |
7 | snssi 4812 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No ) | |
8 | velsn 4646 | . . . 4 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
9 | leftval 27916 | . . . . . . . . . 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 2824 | . . . . . . . 8 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})) |
12 | rabid 3454 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ↔ (𝑥 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝑥 <s 𝐴)) | |
13 | 11, 12 | bitrdi 287 | . . . . . . 7 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) ↔ (𝑥 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝑥 <s 𝐴))) |
14 | 13 | simplbda 499 | . . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘𝐴)) → 𝑥 <s 𝐴) |
15 | breq2 5151 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴)) | |
16 | 14, 15 | imbitrrid 246 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘𝐴)) → 𝑥 <s 𝑦)) |
17 | 16 | expd 415 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) → 𝑥 <s 𝑦))) |
18 | 8, 17 | sylbi 217 | . . 3 ⊢ (𝑦 ∈ {𝐴} → (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) → 𝑥 <s 𝑦))) |
19 | 18 | 3imp231 1112 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘𝐴) ∧ 𝑦 ∈ {𝐴}) → 𝑥 <s 𝑦) |
20 | 1, 3, 6, 7, 19 | ssltd 27850 | 1 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 𝒫 cpw 4604 {csn 4630 class class class wbr 5147 ‘cfv 6562 No csur 27698 <s cslt 27699 bday cbday 27700 <<s csslt 27839 O cold 27896 L cleft 27898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-1o 8504 df-2o 8505 df-no 27701 df-slt 27702 df-bday 27703 df-sslt 27840 df-scut 27842 df-made 27900 df-old 27901 df-left 27903 |
This theorem is referenced by: lltropt 27925 madebdaylemlrcut 27951 mulsproplem5 28160 mulsproplem6 28161 mulsproplem7 28162 mulsproplem8 28163 mulsuniflem 28189 |
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