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Mirrors > Home > MPE Home > Th. List > Mathboxes > lrold | Structured version Visualization version GIF version |
Description: The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 12-Aug-2024.) |
Ref | Expression |
---|---|
lrold | ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leftval 33676 | . . . 4 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}) | |
2 | rightval 33677 | . . . 4 ⊢ (𝐴 ∈ No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}) | |
3 | 1, 2 | uneq12d 4052 | . . 3 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥})) |
4 | unrab 4192 | . . 3 ⊢ ({𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} | |
5 | 3, 4 | eqtrdi 2789 | . 2 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)}) |
6 | oldirr 33702 | . . . . . . 7 ⊢ ¬ 𝐴 ∈ ( O ‘( bday ‘𝐴)) | |
7 | eleq1 2820 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐴 ∈ ( O ‘( bday ‘𝐴)))) | |
8 | 6, 7 | mtbiri 330 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ ( O ‘( bday ‘𝐴))) |
9 | 8 | necon2ai 2963 | . . . . 5 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ≠ 𝐴) |
10 | 9 | adantl 485 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → 𝑥 ≠ 𝐴) |
11 | bdayelon 33604 | . . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On | |
12 | oldf 33674 | . . . . . . . . 9 ⊢ O :On⟶𝒫 No | |
13 | 12 | ffvelrni 6854 | . . . . . . . 8 ⊢ (( bday ‘𝐴) ∈ On → ( O ‘( bday ‘𝐴)) ∈ 𝒫 No ) |
14 | 13 | elpwid 4496 | . . . . . . 7 ⊢ (( bday ‘𝐴) ∈ On → ( O ‘( bday ‘𝐴)) ⊆ No ) |
15 | 11, 14 | ax-mp 5 | . . . . . 6 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No |
16 | 15 | sseli 3871 | . . . . 5 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ∈ No ) |
17 | slttrine 33587 | . . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝐴 ∈ No ) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥))) | |
18 | 17 | ancoms 462 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥))) |
19 | 16, 18 | sylan2 596 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥))) |
20 | 10, 19 | mpbid 235 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)) |
21 | 20 | rabeqcda 3394 | . 2 ⊢ (𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = ( O ‘( bday ‘𝐴))) |
22 | 5, 21 | eqtrd 2773 | 1 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 {crab 3057 ∪ cun 3839 ⊆ wss 3841 𝒫 cpw 4485 class class class wbr 5027 Oncon0 6166 ‘cfv 6333 No csur 33476 <s cslt 33477 bday cbday 33478 O cold 33660 L cleft 33662 R cright 33663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-wrecs 7969 df-recs 8030 df-1o 8124 df-2o 8125 df-no 33479 df-slt 33480 df-bday 33481 df-sslt 33609 df-scut 33611 df-made 33664 df-old 33665 df-left 33667 df-right 33668 |
This theorem is referenced by: lruneq 33716 lrrecval2 33726 |
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