Theorem lrold 28292

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 28244	. . . . 5 ⊢ (L‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴}
2		rightval 28245	. . . . 5 ⊢ (R‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥}
3	1, 2	uneq12i 4164	. . . 4 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = ({𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥})
4		unrab 4259	. . . 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 2783	. . 3 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)}
6		oldirr 28285	. . . . . . . 8 ⊢ ¬ 𝐴 ∈ (O‘(bday‘𝐴))
7		eleq1 2848	. . . . . . . 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 2984	. . . . . 6 ⊢ (𝑥 ∈ (O‘(bday‘𝐴)) → 𝑥 ≠ 𝐴)
10	9	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (O‘(bday‘𝐴))) → 𝑥 ≠ 𝐴)
11		oldno 28239	. . . . . 6 ⊢ (𝑥 ∈ (O‘(bday‘𝐴)) → 𝑥 ∈ No)
12		ltstrine 28117	. . . . . . 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 3423	. . 3 ⊢ (𝐴 ∈ No → {𝑥 ∈ (O‘(bday‘𝐴)) ∣ (𝑥 <s 𝐴 ∨ 𝐴 <s 𝑥)} = (O‘(bday‘𝐴)))
17	5, 16	eqtrid 2807	. 2 ⊢ (𝐴 ∈ No → ((L‘𝐴) ∪ (R‘𝐴)) = (O‘(bday‘𝐴)))
18		un0 4343	. . 3 ⊢ (∅ ∪ ∅) = ∅
19		leftf 28250	. . . . . . 7 ⊢ L:No⟶𝒫 No
20	19	fdmi 6726	. . . . . 6 ⊢ dom L = No
21	20	eleq2i 2852	. . . . 5 ⊢ (𝐴 ∈ dom L ↔ 𝐴 ∈ No)
22		ndmfv 6922	. . . . 5 ⊢ (¬ 𝐴 ∈ dom L → (L‘𝐴) = ∅)
23	21, 22	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ No → (L‘𝐴) = ∅)
24		rightf 28251	. . . . . . 7 ⊢ R:No⟶𝒫 No
25	24	fdmi 6726	. . . . . 6 ⊢ dom R = No
26	25	eleq2i 2852	. . . . 5 ⊢ (𝐴 ∈ dom R ↔ 𝐴 ∈ No)
27		ndmfv 6922	. . . . 5 ⊢ (¬ 𝐴 ∈ dom R → (R‘𝐴) = ∅)
28	26, 27	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ No → (R‘𝐴) = ∅)
29	23, 28	uneq12d 4167	. . 3 ⊢ (¬ 𝐴 ∈ No → ((L‘𝐴) ∪ (R‘𝐴)) = (∅ ∪ ∅))
30		bdaydm 28144	. . . . . . 7 ⊢ dom bday = No
31	30	eleq2i 2852	. . . . . 6 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No)
32		ndmfv 6922	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom bday → (bday‘𝐴) = ∅)
33	31, 32	sylnbir 334	. . . . 5 ⊢ (¬ 𝐴 ∈ No → (bday‘𝐴) = ∅)
34	33	fveq2d 6894	. . . 4 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = (O‘∅))
35		old0 28234	. . . 4 ⊢ (O‘∅) = ∅
36	34, 35	eqtrdi 2811	. . 3 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = ∅)
37	18, 29, 36	3eqtr4a 2821	. 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 1569   ∈ wcel 2144   ≠ wne 2955  {crab 3412   ∪ cun 4148  ∅c0 4277  𝒫 cpw 4556   class class class wbr 5102  dom cdm 5651  ‘cfv 6543  Nocsur 28006   <s clts 28007  bdaycbday 28008  Ocold 28218  Lcleft 28220  Rcright 28221

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-1o 8483  df-2o 8484  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-made 28222  df-old 28223  df-left 28225  df-right 28226

This theorem is used by:  lruneq  28302  bdayiun  28310  lrrecval2  28335  addbdaylem  28412  negbdaylem  28451  onleft  28655  ltonold  28656  oncutlt  28659