Theorem lrold 27906

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 27858	. . . . 5 ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}
2		rightval 27859	. . . . 5 ⊢ ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
3	1, 2	uneq12i 4107	. . . 4 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥})
4		unrab 4256	. . . 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 2760	. . 3 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)}
6		oldirr 27899	. . . . . . . 8 ⊢ ¬ 𝐴 ∈ ( O ‘( bday ‘𝐴))
7		eleq1 2825	. . . . . . . 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 2962	. . . . . 6 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ≠ 𝐴)
10	9	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → 𝑥 ≠ 𝐴)
11		oldno 27853	. . . . . 6 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → 𝑥 ∈ No )
12		ltstrine 27732	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 𝐴 ∈ No ) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
13	12	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
14	11, 13	sylan2 594	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
15	10, 14	mpbid 232	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥))
16	15	rabeqcda 3401	. . 3 ⊢ (𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = ( O ‘( bday ‘𝐴)))
17	5, 16	eqtrid 2784	. 2 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))
18		un0 4335	. . 3 ⊢ (∅ ∪ ∅) = ∅
19		leftf 27864	. . . . . . 7 ⊢ L : No ⟶𝒫 No
20	19	fdmi 6674	. . . . . 6 ⊢ dom L = No
21	20	eleq2i 2829	. . . . 5 ⊢ (𝐴 ∈ dom L ↔ 𝐴 ∈ No )
22		ndmfv 6867	. . . . 5 ⊢ (¬ 𝐴 ∈ dom L → ( L ‘𝐴) = ∅)
23	21, 22	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅)
24		rightf 27865	. . . . . . 7 ⊢ R : No ⟶𝒫 No
25	24	fdmi 6674	. . . . . 6 ⊢ dom R = No
26	25	eleq2i 2829	. . . . 5 ⊢ (𝐴 ∈ dom R ↔ 𝐴 ∈ No )
27		ndmfv 6867	. . . . 5 ⊢ (¬ 𝐴 ∈ dom R → ( R ‘𝐴) = ∅)
28	26, 27	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( R ‘𝐴) = ∅)
29	23, 28	uneq12d 4110	. . 3 ⊢ (¬ 𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (∅ ∪ ∅))
30		bdaydm 27759	. . . . . . 7 ⊢ dom bday = No
31	30	eleq2i 2829	. . . . . 6 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No )
32		ndmfv 6867	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅)
33	31, 32	sylnbir 331	. . . . 5 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅)
34	33	fveq2d 6839	. . . 4 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅))
35		old0 27848	. . . 4 ⊢ ( O ‘∅) = ∅
36	34, 35	eqtrdi 2788	. . 3 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅)
37	18, 29, 36	3eqtr4a 2798	. 2 ⊢ (¬ 𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)))
38	17, 37	pm2.61i 182	1 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3390   ∪ cun 3888  ∅c0 4274  𝒫 cpw 4542   class class class wbr 5086  dom cdm 5625  ‘cfv 6493   No csur 27620   <s clts 27621   bday cbday 27622   O cold 27832   L cleft 27834   R cright 27835

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27623  df-lts 27624  df-bday 27625  df-slts 27767  df-cuts 27769  df-made 27836  df-old 27837  df-left 27839  df-right 27840

This theorem is referenced by:  lruneq  27916  bdayiun  27924  lrrecval2  27949  addbdaylem  28026  negbdaylem  28065  onleft  28269  ltonold  28270  oncutlt  28273