Theorem ltonold 28677

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 28168	. . . . . . 7 ⊢ (bday‘𝑥) ∈ On
2	1	onordi 6494	. . . . . 6 ⊢ Ord (bday‘𝑥)
3		bdayon 28168	. . . . . . 7 ⊢ (bday‘𝐴) ∈ On
4	3	onordi 6494	. . . . . 6 ⊢ Ord (bday‘𝐴)
5		ordtri2or 6481	. . . . . 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 28300	. . . . . . . . . 10 ⊢ (M‘(bday‘𝑥)) = ((O‘(bday‘𝑥)) ∪ (N‘(bday‘𝑥)))
9	8	eleq2i 2856	. . . . . . . . 9 ⊢ (𝐴 ∈ (M‘(bday‘𝑥)) ↔ 𝐴 ∈ ((O‘(bday‘𝑥)) ∪ (N‘(bday‘𝑥))))
10		elun 4155	. . . . . . . . 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 28313	. . . . . . . . . . 11 ⊢ ((L‘𝑥) ∪ (R‘𝑥)) = (O‘(bday‘𝑥))
13	12	eleq2i 2856	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((L‘𝑥) ∪ (R‘𝑥)) ↔ 𝐴 ∈ (O‘(bday‘𝑥)))
14		elons 28669	. . . . . . . . . . . . . . . 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 4170	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((L‘𝑥) ∪ (R‘𝑥)) = ((L‘𝑥) ∪ ∅))
18		un0 4347	. . . . . . . . . . . . 13 ⊢ ((L‘𝑥) ∪ ∅) = (L‘𝑥)
19	17, 18	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((L‘𝑥) ∪ (R‘𝑥)) = (L‘𝑥))
20	19	eleq2d 2850	. . . . . . . . . . 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 28671	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Ons → 𝑥 ∈ No)
23	22	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ (L‘𝑥)) → 𝑥 ∈ No)
24		leftlt 28269	. . . . . . . . . . . . . 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 28160	. . . . . . . . . . . 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 28318	. . . . . . . . . . . 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 28287	. . . . . . . . . . . . 13 ⊢ (L‘𝐴) ⊆ (O‘(bday‘𝐴))
34		fveq2 6908	. . . . . . . . . . . . . . 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 28676	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ (bday‘𝐴) = (bday‘𝑥)) → (L‘𝑥) = (O‘(bday‘𝐴)))
39	33, 38	sseqtrrid 3956	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ (bday‘𝐴) = (bday‘𝑥)) → (L‘𝐴) ⊆ (L‘𝑥))
40		leslss 28325	. . . . . . . . . . . . . 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 28316	. . . . . . . . 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 28139	. . . . . . . 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 28317	. . . 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 4002	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 1570   ∈ wcel 2147  {crab 3416   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281   class class class wbr 5106  Ord word 6379  Oncon0 6380  ‘cfv 6558  Nocsur 28027   <s clts 28028  bdaycbday 28029   ≤s cles 28131  Mcmade 28238  Ocold 28239  Ncnew 28240  Lcleft 28241  Rcright 28242  On_scons 28667

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-made 28243  df-old 28244  df-new 28245  df-left 28246  df-right 28247  df-ons 28668

This theorem is used by:  ltonsex  28678  onsfi  28772