Theorem lrold 27950

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 27917	. . . . 5 ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}
2		rightval 27918	. . . . 5 ⊢ ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
3	1, 2	uneq12i 4176	. . . 4 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥})
4		unrab 4321	. . . 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 2763	. . 3 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)}
6		oldirr 27943	. . . . . . . 8 ⊢ ¬ 𝐴 ∈ ( O ‘( bday ‘𝐴))
7		eleq1 2827	. . . . . . . 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 2968	. . . . . 6 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ≠ 𝐴)
10	9	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → 𝑥 ≠ 𝐴)
11		oldssno 27915	. . . . . . 7 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
12	11	sseli 3991	. . . . . 6 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ∈ No )
13		slttrine 27811	. . . . . . 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 3445	. . 3 ⊢ (𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = ( O ‘( bday ‘𝐴)))
18	5, 17	eqtrid 2787	. 2 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))
19		un0 4400	. . 3 ⊢ (∅ ∪ ∅) = ∅
20		leftf 27919	. . . . . . 7 ⊢ L : No ⟶𝒫 No
21	20	fdmi 6748	. . . . . 6 ⊢ dom L = No
22	21	eleq2i 2831	. . . . 5 ⊢ (𝐴 ∈ dom L ↔ 𝐴 ∈ No )
23		ndmfv 6942	. . . . 5 ⊢ (¬ 𝐴 ∈ dom L → ( L ‘𝐴) = ∅)
24	22, 23	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅)
25		rightf 27920	. . . . . . 7 ⊢ R : No ⟶𝒫 No
26	25	fdmi 6748	. . . . . 6 ⊢ dom R = No
27	26	eleq2i 2831	. . . . 5 ⊢ (𝐴 ∈ dom R ↔ 𝐴 ∈ No )
28		ndmfv 6942	. . . . 5 ⊢ (¬ 𝐴 ∈ dom R → ( R ‘𝐴) = ∅)
29	27, 28	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( R ‘𝐴) = ∅)
30	24, 29	uneq12d 4179	. . 3 ⊢ (¬ 𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (∅ ∪ ∅))
31		bdaydm 27834	. . . . . . 7 ⊢ dom bday = No
32	31	eleq2i 2831	. . . . . 6 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No )
33		ndmfv 6942	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅)
34	32, 33	sylnbir 331	. . . . 5 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅)
35	34	fveq2d 6911	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅))
36		old0 27913	. . . 4 ⊢ ( O ‘∅) = ∅
37	35, 36	eqtrdi 2791	. . 3 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅)
38	19, 30, 37	3eqtr4a 2801	. 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 1537   ∈ wcel 2106   ≠ wne 2938  {crab 3433   ∪ cun 3961  ∅c0 4339  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689  ‘cfv 6563   No csur 27699   <s cslt 27700   bday cbday 27701   O cold 27897   L cleft 27899   R cright 27900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-left 27904  df-right 27905

This theorem is referenced by:  lruneq  27959  lrrecval2  27988  addsbdaylem  28064  negsbdaylem  28103  sltonold  28298