Theorem lrold 27815

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 27778	. . . . 5 ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}
2		rightval 27779	. . . . 5 ⊢ ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
3	1, 2	uneq12i 4132	. . . 4 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥})
4		unrab 4281	. . . 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 2753	. . 3 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)}
6		oldirr 27808	. . . . . . . 8 ⊢ ¬ 𝐴 ∈ ( O ‘( bday ‘𝐴))
7		eleq1 2817	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐴 ∈ ( O ‘( bday ‘𝐴))))
8	6, 7	mtbiri 327	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ ( O ‘( bday ‘𝐴)))
9	8	necon2ai 2955	. . . . . 6 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ≠ 𝐴)
10	9	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → 𝑥 ≠ 𝐴)
11		oldssno 27776	. . . . . . 7 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
12	11	sseli 3945	. . . . . 6 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ∈ No )
13		slttrine 27670	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 𝐴 ∈ No ) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
14	13	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
15	12, 14	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
16	10, 15	mpbid 232	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥))
17	16	rabeqcda 3420	. . 3 ⊢ (𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = ( O ‘( bday ‘𝐴)))
18	5, 17	eqtrid 2777	. 2 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))
19		un0 4360	. . 3 ⊢ (∅ ∪ ∅) = ∅
20		leftf 27784	. . . . . . 7 ⊢ L : No ⟶𝒫 No
21	20	fdmi 6702	. . . . . 6 ⊢ dom L = No
22	21	eleq2i 2821	. . . . 5 ⊢ (𝐴 ∈ dom L ↔ 𝐴 ∈ No )
23		ndmfv 6896	. . . . 5 ⊢ (¬ 𝐴 ∈ dom L → ( L ‘𝐴) = ∅)
24	22, 23	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅)
25		rightf 27785	. . . . . . 7 ⊢ R : No ⟶𝒫 No
26	25	fdmi 6702	. . . . . 6 ⊢ dom R = No
27	26	eleq2i 2821	. . . . 5 ⊢ (𝐴 ∈ dom R ↔ 𝐴 ∈ No )
28		ndmfv 6896	. . . . 5 ⊢ (¬ 𝐴 ∈ dom R → ( R ‘𝐴) = ∅)
29	27, 28	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( R ‘𝐴) = ∅)
30	24, 29	uneq12d 4135	. . 3 ⊢ (¬ 𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (∅ ∪ ∅))
31		bdaydm 27693	. . . . . . 7 ⊢ dom bday = No
32	31	eleq2i 2821	. . . . . 6 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No )
33		ndmfv 6896	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅)
34	32, 33	sylnbir 331	. . . . 5 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅)
35	34	fveq2d 6865	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅))
36		old0 27774	. . . 4 ⊢ ( O ‘∅) = ∅
37	35, 36	eqtrdi 2781	. . 3 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅)
38	19, 30, 37	3eqtr4a 2791	. 2 ⊢ (¬ 𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))
39	18, 38	pm2.61i 182	1 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  {crab 3408   ∪ cun 3915  ∅c0 4299  𝒫 cpw 4566   class class class wbr 5110  dom cdm 5641  ‘cfv 6514   No csur 27558   <s cslt 27559   bday cbday 27560   O cold 27758   L cleft 27760   R cright 27761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-made 27762  df-old 27763  df-left 27765  df-right 27766

This theorem is referenced by:  lruneq  27825  lrrecval2  27854  addsbdaylem  27930  negsbdaylem  27969  onsleft  28168  sltonold  28169  onscutlt  28172