Theorem lrold 28283

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 28235	. . . . 5 ⊢ (L‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴}
2		rightval 28236	. . . . 5 ⊢ (R‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥}
3	1, 2	uneq12i 4113	. . . 4 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = ({𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥})
4		unrab 4261	. . . 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 2784	. . 3 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)}
6		oldirr 28276	. . . . . . . 8 ⊢ ¬ 𝐴 ∈ (O‘(bday‘𝐴))
7		eleq1 2849	. . . . . . . 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 2985	. . . . . 6 ⊢ (𝑥 ∈ (O‘(bday‘𝐴)) → 𝑥 ≠ 𝐴)
10	9	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (O‘(bday‘𝐴))) → 𝑥 ≠ 𝐴)
11		oldno 28230	. . . . . 6 ⊢ (𝑥 ∈ (O‘(bday‘𝐴)) → 𝑥 ∈ No)
12		ltstrine 28108	. . . . . . 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 3424	. . 3 ⊢ (𝐴 ∈ No → {𝑥 ∈ (O‘(bday‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = (O‘(bday‘𝐴)))
17	5, 16	eqtrid 2808	. 2 ⊢ (𝐴 ∈ No → ((L‘𝐴) ∪ (R‘𝐴)) = (O‘(bday‘𝐴)))
18		un0 4344	. . 3 ⊢ (∅ ∪ ∅) = ∅
19		leftf 28241	. . . . . . 7 ⊢ L:No⟶𝒫 No
20	19	fdmi 6721	. . . . . 6 ⊢ dom L = No
21	20	eleq2i 2853	. . . . 5 ⊢ (𝐴 ∈ dom L ↔ 𝐴 ∈ No)
22		ndmfv 6917	. . . . 5 ⊢ (¬ 𝐴 ∈ dom L → (L‘𝐴) = ∅)
23	21, 22	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ No → (L‘𝐴) = ∅)
24		rightf 28242	. . . . . . 7 ⊢ R:No⟶𝒫 No
25	24	fdmi 6721	. . . . . 6 ⊢ dom R = No
26	25	eleq2i 2853	. . . . 5 ⊢ (𝐴 ∈ dom R ↔ 𝐴 ∈ No)
27		ndmfv 6917	. . . . 5 ⊢ (¬ 𝐴 ∈ dom R → (R‘𝐴) = ∅)
28	26, 27	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ No → (R‘𝐴) = ∅)
29	23, 28	uneq12d 4116	. . 3 ⊢ (¬ 𝐴 ∈ No → ((L‘𝐴) ∪ (R‘𝐴)) = (∅ ∪ ∅))
30		bdaydm 28135	. . . . . . 7 ⊢ dom bday = No
31	30	eleq2i 2853	. . . . . 6 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No)
32		ndmfv 6917	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom bday → (bday‘𝐴) = ∅)
33	31, 32	sylnbir 334	. . . . 5 ⊢ (¬ 𝐴 ∈ No → (bday‘𝐴) = ∅)
34	33	fveq2d 6889	. . . 4 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = (O‘∅))
35		old0 28225	. . . 4 ⊢ (O‘∅) = ∅
36	34, 35	eqtrdi 2812	. . 3 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = ∅)
37	18, 29, 36	3eqtr4a 2822	. 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 2145   ≠ wne 2956  {crab 3413   ∪ cun 3897  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103  dom cdm 5651  ‘cfv 6538  Nocsur 27997   <s clts 27998  bdaycbday 27999  Ocold 28209  Lcleft 28211  Rcright 28212

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217

This theorem is used by:  lruneq  28293  bdayiun  28301  lrrecval2  28326  addbdaylem  28403  negbdaylem  28442  onleft  28646  ltonold  28647  oncutlt  28650