Theorem ltonold 28652

Description: The class of ordinals less than any surreal is a subset of that surreal's old set. (Contributed by Scott Fenton, 22-Mar-2025.)

Assertion

Ref	Expression
ltonold	⊢ (𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ (O‘(bday‘𝐴)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem ltonold

Step	Hyp	Ref	Expression
1		bdayon 28143	. . . . . . 7 ⊢ (bday‘𝑥) ∈ On
2	1	onordi 6476	. . . . . 6 ⊢ Ord (bday‘𝑥)
3		bdayon 28143	. . . . . . 7 ⊢ (bday‘𝐴) ∈ On
4	3	onordi 6476	. . . . . 6 ⊢ Ord (bday‘𝐴)
5		ordtri2or 6463	. . . . . 6 ⊢ ((Ord (bday‘𝑥) ∧ Ord (bday‘𝐴)) → ((bday‘𝑥) ∈ (bday‘𝐴) ∨ (bday‘𝐴) ⊆ (bday‘𝑥)))
6	2, 4, 5	mp2an 705	. . . . 5 ⊢ ((bday‘𝑥) ∈ (bday‘𝐴) ∨ (bday‘𝐴) ⊆ (bday‘𝑥))
7	6	a1i 11	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → ((bday‘𝑥) ∈ (bday‘𝐴) ∨ (bday‘𝐴) ⊆ (bday‘𝑥)))
8		madeun 28275	. . . . . . . . . 10 ⊢ (M‘(bday‘𝑥)) = ((O‘(bday‘𝑥)) ∪ (N‘(bday‘𝑥)))
9	8	eleq2i 2852	. . . . . . . . 9 ⊢ (𝐴 ∈ (M‘(bday‘𝑥)) ↔ 𝐴 ∈ ((O‘(bday‘𝑥)) ∪ (N‘(bday‘𝑥))))
10		elun 4152	. . . . . . . . 9 ⊢ (𝐴 ∈ ((O‘(bday‘𝑥)) ∪ (N‘(bday‘𝑥))) ↔ (𝐴 ∈ (O‘(bday‘𝑥)) ∨ 𝐴 ∈ (N‘(bday‘𝑥))))
11	9, 10	bitri 278	. . . . . . . 8 ⊢ (𝐴 ∈ (M‘(bday‘𝑥)) ↔ (𝐴 ∈ (O‘(bday‘𝑥)) ∨ 𝐴 ∈ (N‘(bday‘𝑥))))
12		lrold 28288	. . . . . . . . . . 11 ⊢ ((L‘𝑥) ∪ (R‘𝑥)) = (O‘(bday‘𝑥))
13	12	eleq2i 2852	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((L‘𝑥) ∪ (R‘𝑥)) ↔ 𝐴 ∈ (O‘(bday‘𝑥)))
14		elons 28644	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Ons ↔ (𝑥 ∈ No ∧ (R‘𝑥) = ∅))
15	14	simprbi 503	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Ons → (R‘𝑥) = ∅)
16	15	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (R‘𝑥) = ∅)
17	16	uneq2d 4167	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((L‘𝑥) ∪ (R‘𝑥)) = ((L‘𝑥) ∪ ∅))
18		un0 4343	. . . . . . . . . . . . 13 ⊢ ((L‘𝑥) ∪ ∅) = (L‘𝑥)
19	17, 18	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((L‘𝑥) ∪ (R‘𝑥)) = (L‘𝑥))
20	19	eleq2d 2846	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ((L‘𝑥) ∪ (R‘𝑥)) ↔ 𝐴 ∈ (L‘𝑥)))
21		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ (L‘𝑥)) → 𝐴 ∈ No)
22		onno 28646	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Ons → 𝑥 ∈ No)
23	22	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ (L‘𝑥)) → 𝑥 ∈ No)
24		leftlt 28244	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (L‘𝑥) → 𝐴 <s 𝑥)
25	24	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ (L‘𝑥)) → 𝐴 <s 𝑥)
26	21, 23, 25	ltlesd 28135	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ (L‘𝑥)) → 𝐴 ≤s 𝑥)
27	26	ex 418	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (L‘𝑥) → 𝐴 ≤s 𝑥))
28	20, 27	sylbid 243	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ((L‘𝑥) ∪ (R‘𝑥)) → 𝐴 ≤s 𝑥))
29	13, 28	biimtrrid 246	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (O‘(bday‘𝑥)) → 𝐴 ≤s 𝑥))
30		newbday 28293	. . . . . . . . . . . 12 ⊢ (((bday‘𝑥) ∈ On ∧ 𝐴 ∈ No) → (𝐴 ∈ (N‘(bday‘𝑥)) ↔ (bday‘𝐴) = (bday‘𝑥)))
31	1, 30	mpan 703	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → (𝐴 ∈ (N‘(bday‘𝑥)) ↔ (bday‘𝐴) = (bday‘𝑥)))
32	31	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (N‘(bday‘𝑥)) ↔ (bday‘𝐴) = (bday‘𝑥)))
33		leftssold 28262	. . . . . . . . . . . . 13 ⊢ (L‘𝐴) ⊆ (O‘(bday‘𝐴))
34		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ ((bday‘𝐴) = (bday‘𝑥) → (O‘(bday‘𝐴)) = (O‘(bday‘𝑥)))
35	34	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ (bday‘𝐴) = (bday‘𝑥)) → (O‘(bday‘𝐴)) = (O‘(bday‘𝑥)))
36		onleft 28651	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Ons → (O‘(bday‘𝑥)) = (L‘𝑥))
37	36	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ (bday‘𝐴) = (bday‘𝑥)) → (O‘(bday‘𝑥)) = (L‘𝑥))
38	35, 37	eqtr2d 2796	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ (bday‘𝐴) = (bday‘𝑥)) → (L‘𝑥) = (O‘(bday‘𝐴)))
39	33, 38	sseqtrrid 3951	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ (bday‘𝐴) = (bday‘𝑥)) → (L‘𝐴) ⊆ (L‘𝑥))
40		leslss 28300	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ∧ (bday‘𝐴) = (bday‘𝑥)) → (𝐴 ≤s 𝑥 ↔ (L‘𝐴) ⊆ (L‘𝑥)))
41	22, 40	syl3an2 1182	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝑥)) → (𝐴 ≤s 𝑥 ↔ (L‘𝐴) ⊆ (L‘𝑥)))
42	41	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ (bday‘𝐴) = (bday‘𝑥)) → (𝐴 ≤s 𝑥 ↔ (L‘𝐴) ⊆ (L‘𝑥)))
43	39, 42	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ (bday‘𝐴) = (bday‘𝑥)) → 𝐴 ≤s 𝑥)
44	43	ex 418	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((bday‘𝐴) = (bday‘𝑥) → 𝐴 ≤s 𝑥))
45	32, 44	sylbid 243	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (N‘(bday‘𝑥)) → 𝐴 ≤s 𝑥))
46	29, 45	jaod 873	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((𝐴 ∈ (O‘(bday‘𝑥)) ∨ 𝐴 ∈ (N‘(bday‘𝑥))) → 𝐴 ≤s 𝑥))
47	11, 46	biimtrid 245	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (M‘(bday‘𝑥)) → 𝐴 ≤s 𝑥))
48		madebday 28291	. . . . . . . . 9 ⊢ (((bday‘𝑥) ∈ On ∧ 𝐴 ∈ No) → (𝐴 ∈ (M‘(bday‘𝑥)) ↔ (bday‘𝐴) ⊆ (bday‘𝑥)))
49	1, 48	mpan 703	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴 ∈ (M‘(bday‘𝑥)) ↔ (bday‘𝐴) ⊆ (bday‘𝑥)))
50	49	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (M‘(bday‘𝑥)) ↔ (bday‘𝐴) ⊆ (bday‘𝑥)))
51		lenlts 28114	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No) → (𝐴 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝐴))
52	22, 51	sylan2 605	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝐴))
53	47, 50, 52	3imtr3d 296	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((bday‘𝐴) ⊆ (bday‘𝑥) → ¬ 𝑥 <s 𝐴))
54	53	con2d 135	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝑥 <s 𝐴 → ¬ (bday‘𝐴) ⊆ (bday‘𝑥)))
55	54	3impia 1135	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → ¬ (bday‘𝐴) ⊆ (bday‘𝑥))
56	7, 55	olcnd 891	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → (bday‘𝑥) ∈ (bday‘𝐴))
57	22	3ad2ant2 1152	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → 𝑥 ∈ No)
58		oldbday 28292	. . . 4 ⊢ (((bday‘𝐴) ∈ On ∧ 𝑥 ∈ No) → (𝑥 ∈ (O‘(bday‘𝐴)) ↔ (bday‘𝑥) ∈ (bday‘𝐴)))
59	3, 57, 58	sylancr 599	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → (𝑥 ∈ (O‘(bday‘𝐴)) ↔ (bday‘𝑥) ∈ (bday‘𝐴)))
60	56, 59	mpbird 260	. 2 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → 𝑥 ∈ (O‘(bday‘𝐴)))
61	60	rabssdv 3999	1 ⊢ (𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ (O‘(bday‘𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278   class class class wbr 5102  Ord word 6361  Oncon0 6362  ‘cfv 6538  Nocsur 28002   <s clts 28003  bdaycbday 28004   ≤s cles 28106  Mcmade 28213  Ocold 28214  Ncnew 28215  Lcleft 28216  Rcright 28217  On_scons 28642

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 5326  ax-pr 5390  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 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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-made 28218  df-old 28219  df-new 28220  df-left 28221  df-right 28222  df-ons 28643

This theorem is used by:  ltonsex  28653  onsfi  28747