Theorem lrold 27907

Description: The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)

Assertion

Ref	Expression
lrold	⊢ (( 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 27859	. . . . 5 ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}
2		rightval 27860	. . . . 5 ⊢ ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
3	1, 2	uneq12i 4096	. . . 4 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥})
4		unrab 4243	. . . 4 ⊢ ({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	eqtri 2762	. . 3 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)}
6		oldirr 27900	. . . . . . . 8 ⊢ ¬ 𝐴 ∈ ( O ‘( bday ‘𝐴))
7		eleq1 2827	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐴 ∈ ( O ‘( bday ‘𝐴))))
8	6, 7	mtbiri 328	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ ( O ‘( bday ‘𝐴)))
9	8	necon2ai 2963	. . . . . 6 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ≠ 𝐴)
10	9	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → 𝑥 ≠ 𝐴)
11		oldno 27854	. . . . . 6 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ∈ No )
12		ltstrine 27733	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 𝐴 ∈ No ) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
13	12	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
14	11, 13	sylan2 599	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
15	10, 14	mpbid 233	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥))
16	15	rabeqcda 3402	. . 3 ⊢ (𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = ( O ‘( bday ‘𝐴)))
17	5, 16	eqtrid 2786	. 2 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))
18		un0 4322	. . 3 ⊢ (∅ ∪ ∅) = ∅
19		leftf 27865	. . . . . . 7 ⊢ L : No ⟶𝒫 No
20	19	fdmi 6666	. . . . . 6 ⊢ dom L = No
21	20	eleq2i 2831	. . . . 5 ⊢ (𝐴 ∈ dom L ↔ 𝐴 ∈ No )
22		ndmfv 6859	. . . . 5 ⊢ (¬ 𝐴 ∈ dom L → ( L ‘𝐴) = ∅)
23	21, 22	sylnbir 332	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅)
24		rightf 27866	. . . . . . 7 ⊢ R : No ⟶𝒫 No
25	24	fdmi 6666	. . . . . 6 ⊢ dom R = No
26	25	eleq2i 2831	. . . . 5 ⊢ (𝐴 ∈ dom R ↔ 𝐴 ∈ No )
27		ndmfv 6859	. . . . 5 ⊢ (¬ 𝐴 ∈ dom R → ( R ‘𝐴) = ∅)
28	26, 27	sylnbir 332	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( R ‘𝐴) = ∅)
29	23, 28	uneq12d 4099	. . 3 ⊢ (¬ 𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (∅ ∪ ∅))
30		bdaydm 27760	. . . . . . 7 ⊢ dom bday = No
31	30	eleq2i 2831	. . . . . 6 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No )
32		ndmfv 6859	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅)
33	31, 32	sylnbir 332	. . . . 5 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅)
34	33	fveq2d 6831	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅))
35		old0 27849	. . . 4 ⊢ ( O ‘∅) = ∅
36	34, 35	eqtrdi 2790	. . 3 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅)
37	18, 29, 36	3eqtr4a 2800	. 2 ⊢ (¬ 𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))
38	17, 37	pm2.61i 183	1 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  {crab 3391   ∪ cun 3881  ∅c0 4261  𝒫 cpw 4529   class class class wbr 5072  dom cdm 5618  ‘cfv 6485   No csur 27621   <s clts 27622   bday cbday 27623   O cold 27833   L cleft 27835   R cright 27836

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 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-no 27624  df-lts 27625  df-bday 27626  df-slts 27768  df-cuts 27770  df-made 27837  df-old 27838  df-left 27840  df-right 27841

This theorem is referenced by:  lruneq  27917  bdayiun  27925  lrrecval2  27950  addbdaylem  28027  negbdaylem  28066  onleft  28270  ltonold  28271  oncutlt  28274