Theorem lrold 28340

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 28292	. . . . 5 ⊢ (L‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴}
2		rightval 28293	. . . . 5 ⊢ (R‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥}
3	1, 2	uneq12i 4168	. . . 4 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = ({𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥})
4		unrab 4263	. . . 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 2786	. . 3 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)}
6		oldirr 28333	. . . . . . . 8 ⊢ ¬ 𝐴 ∈ (O‘(bday‘𝐴))
7		eleq1 2852	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (O‘(bday‘𝐴)) ↔ 𝐴 ∈ (O‘(bday‘𝐴))))
8	6, 7	mtbiri 330	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ (O‘(bday‘𝐴)))
9	8	necon2ai 2988	. . . . . 6 ⊢ (𝑥 ∈ (O‘(bday‘𝐴)) → 𝑥 ≠ 𝐴)
10	9	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (O‘(bday‘𝐴))) → 𝑥 ≠ 𝐴)
11		oldno 28287	. . . . . 6 ⊢ (𝑥 ∈ (O‘(bday‘𝐴)) → 𝑥 ∈ No)
12		ltstrine 28165	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ 𝐴 ∈ No) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
13	12	ancoms 464	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
14	11, 13	sylan2 605	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (O‘(bday‘𝐴))) → (𝑥 ≠ 𝐴 ↔ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)))
15	10, 14	mpbid 235	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (O‘(bday‘𝐴))) → (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥))
16	15	rabeqcda 3427	. . 3 ⊢ (𝐴 ∈ No → {𝑥 ∈ (O‘(bday‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = (O‘(bday‘𝐴)))
17	5, 16	eqtrid 2810	. 2 ⊢ (𝐴 ∈ No → ((L‘𝐴) ∪ (R‘𝐴)) = (O‘(bday‘𝐴)))
18		un0 4347	. . 3 ⊢ (∅ ∪ ∅) = ∅
19		leftf 28298	. . . . . . 7 ⊢ L:No⟶𝒫 No
20	19	fdmi 6751	. . . . . 6 ⊢ dom L = No
21	20	eleq2i 2856	. . . . 5 ⊢ (𝐴 ∈ dom L ↔ 𝐴 ∈ No)
22		ndmfv 6948	. . . . 5 ⊢ (¬ 𝐴 ∈ dom L → (L‘𝐴) = ∅)
23	21, 22	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ No → (L‘𝐴) = ∅)
24		rightf 28299	. . . . . . 7 ⊢ R:No⟶𝒫 No
25	24	fdmi 6751	. . . . . 6 ⊢ dom R = No
26	25	eleq2i 2856	. . . . 5 ⊢ (𝐴 ∈ dom R ↔ 𝐴 ∈ No)
27		ndmfv 6948	. . . . 5 ⊢ (¬ 𝐴 ∈ dom R → (R‘𝐴) = ∅)
28	26, 27	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ No → (R‘𝐴) = ∅)
29	23, 28	uneq12d 4171	. . 3 ⊢ (¬ 𝐴 ∈ No → ((L‘𝐴) ∪ (R‘𝐴)) = (∅ ∪ ∅))
30		bdaydm 28192	. . . . . . 7 ⊢ dom bday = No
31	30	eleq2i 2856	. . . . . 6 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No)
32		ndmfv 6948	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom bday → (bday‘𝐴) = ∅)
33	31, 32	sylnbir 334	. . . . 5 ⊢ (¬ 𝐴 ∈ No → (bday‘𝐴) = ∅)
34	33	fveq2d 6920	. . . 4 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = (O‘∅))
35		old0 28282	. . . 4 ⊢ (O‘∅) = ∅
36	34, 35	eqtrdi 2814	. . 3 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = ∅)
37	18, 29, 36	3eqtr4a 2824	. 2 ⊢ (¬ 𝐴 ∈ No → ((L‘𝐴) ∪ (R‘𝐴)) = (O‘(bday‘𝐴)))
38	17, 37	pm2.61i 184	1 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = (O‘(bday‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  Nocsur 28054   <s clts 28055  bdaycbday 28056  Ocold 28266  Lcleft 28268  Rcright 28269

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270  df-old 28271  df-left 28273  df-right 28274

This theorem is used by:  lruneq  28350  bdayiun  28358  lrrecval2  28383  addbdaylem  28460  negbdaylem  28499  onleft  28703  ltonold  28704  oncutlt  28707