Theorem lrold 33660

Description: The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 12-Aug-2024.)

Assertion

Ref	Expression
lrold	⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))

Proof of Theorem lrold

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		leftval 33629	. . . 4 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})
2		rightval 33630	. . . 4 ⊢ (𝐴 ∈ No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥})
3	1, 2	uneq12d 4071	. . 3 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}))
4		unrab 4210	. . 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 2809	. 2 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)})
6		oldirr 33655	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ( O ‘( bday ‘𝐴))
7		eleq1 2839	. . . . . . 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 2980	. . . . 5 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ≠ 𝐴)
10	9	adantl 485	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → 𝑥 ≠ 𝐴)
11		bdayelon 33560	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
12		oldf 33627	. . . . . . . . 9 ⊢ O :On⟶𝒫 No
13	12	ffvelrni 6846	. . . . . . . 8 ⊢ (( bday ‘𝐴) ∈ On → ( O ‘( bday ‘𝐴)) ∈ 𝒫 No )
14	13	elpwid 4508	. . . . . . 7 ⊢ (( bday ‘𝐴) ∈ On → ( O ‘( bday ‘𝐴)) ⊆ No )
15	11, 14	ax-mp 5	. . . . . 6 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
16	15	sseli 3890	. . . . 5 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ∈ No )
17		slttrine 33543	. . . . . 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 595	. . . 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 3406	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = ( O ‘( bday ‘𝐴)))
22	5, 21	eqtrd 2793	1 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  {crab 3074   ∪ cun 3858   ⊆ wss 3860  𝒫 cpw 4497   class class class wbr 5035  Oncon0 6173  ‘cfv 6339   No csur 33432   <s cslt 33433   bday cbday 33434   O cold 33613   L cleft 33615   R cright 33616

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 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464

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 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-wrecs 7962  df-recs 8023  df-1o 8117  df-2o 8118  df-no 33435  df-slt 33436  df-bday 33437  df-sslt 33565  df-scut 33567  df-made 33617  df-old 33618  df-left 33620  df-right 33621

This theorem is referenced by:  lrrecval2  33671