Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 | leftf 33639 | . . . 4 ⊢ L : No ⟶𝒫 No | |
2 | 1 | ffvelrni 6847 | . . 3 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ∈ 𝒫 No ) |
3 | 2 | elpwid 4508 | . 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No ) |
4 | snssi 4701 | . 2 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No ) | |
5 | leftval 33637 | . . . . . 6 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑦 <s 𝐴}) | |
6 | 5 | eleq2d 2837 | . . . . 5 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) ↔ 𝑥 ∈ {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑦 <s 𝐴})) |
7 | breq1 5039 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 <s 𝐴 ↔ 𝑥 <s 𝐴)) | |
8 | 7 | elrab 3604 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑦 <s 𝐴} ↔ (𝑥 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝑥 <s 𝐴)) |
9 | 8 | simprbi 500 | . . . . 5 ⊢ (𝑥 ∈ {𝑦 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑦 <s 𝐴} → 𝑥 <s 𝐴) |
10 | 6, 9 | syl6bi 256 | . . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘𝐴) → 𝑥 <s 𝐴)) |
11 | 10 | ralrimiv 3112 | . . 3 ⊢ (𝐴 ∈ No → ∀𝑥 ∈ ( L ‘𝐴)𝑥 <s 𝐴) |
12 | breq2 5040 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴)) | |
13 | 12 | ralsng 4573 | . . . 4 ⊢ (𝐴 ∈ No → (∀𝑦 ∈ {𝐴}𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴)) |
14 | 13 | ralbidv 3126 | . . 3 ⊢ (𝐴 ∈ No → (∀𝑥 ∈ ( L ‘𝐴)∀𝑦 ∈ {𝐴}𝑥 <s 𝑦 ↔ ∀𝑥 ∈ ( L ‘𝐴)𝑥 <s 𝐴)) |
15 | 11, 14 | mpbird 260 | . 2 ⊢ (𝐴 ∈ No → ∀𝑥 ∈ ( L ‘𝐴)∀𝑦 ∈ {𝐴}𝑥 <s 𝑦) |
16 | fvex 6676 | . . . 4 ⊢ ( L ‘𝐴) ∈ V | |
17 | snex 5304 | . . . 4 ⊢ {𝐴} ∈ V | |
18 | 16, 17 | pm3.2i 474 | . . 3 ⊢ (( L ‘𝐴) ∈ V ∧ {𝐴} ∈ V) |
19 | brsslt 33577 | . . 3 ⊢ (( L ‘𝐴) <<s {𝐴} ↔ ((( L ‘𝐴) ∈ V ∧ {𝐴} ∈ V) ∧ (( L ‘𝐴) ⊆ No ∧ {𝐴} ⊆ No ∧ ∀𝑥 ∈ ( L ‘𝐴)∀𝑦 ∈ {𝐴}𝑥 <s 𝑦))) | |
20 | 18, 19 | mpbiran 708 | . 2 ⊢ (( L ‘𝐴) <<s {𝐴} ↔ (( L ‘𝐴) ⊆ No ∧ {𝐴} ⊆ No ∧ ∀𝑥 ∈ ( L ‘𝐴)∀𝑦 ∈ {𝐴}𝑥 <s 𝑦)) |
21 | 3, 4, 15, 20 | syl3anbrc 1340 | 1 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ∀wral 3070 {crab 3074 Vcvv 3409 ⊆ wss 3860 𝒫 cpw 4497 {csn 4525 class class class wbr 5036 ‘cfv 6340 No csur 33440 <s cslt 33441 bday cbday 33442 <<s csslt 33572 O cold 33621 L cleft 33623 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-wrecs 7963 df-recs 8024 df-1o 8118 df-2o 8119 df-no 33443 df-slt 33444 df-bday 33445 df-sslt 33573 df-scut 33575 df-made 33625 df-old 33626 df-left 33628 |
This theorem is referenced by: lltropt 33646 madebdaylemlrcut 33670 |
Copyright terms: Public domain | W3C validator |