Theorem sltonold 28301

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem sltonold

Dummy variable 𝑥_𝑂 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayelon 27839	. . . . . . 7 ⊢ ( bday ‘𝑥) ∈ On
2	1	onordi 6506	. . . . . 6 ⊢ Ord ( bday ‘𝑥)
3		bdayelon 27839	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
4	3	onordi 6506	. . . . . 6 ⊢ Ord ( bday ‘𝐴)
5		ordtri2or 6493	. . . . . 6 ⊢ ((Ord ( bday ‘𝑥) ∧ Ord ( bday ‘𝐴)) → (( bday ‘𝑥) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
6	2, 4, 5	mp2an 691	. . . . 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 27940	. . . . . . . . . 10 ⊢ ( M ‘( bday ‘𝑥)) = (( O ‘( bday ‘𝑥)) ∪ ( N ‘( bday ‘𝑥)))
9	8	eleq2i 2836	. . . . . . . . 9 ⊢ (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝐴 ∈ (( O ‘( bday ‘𝑥)) ∪ ( N ‘( bday ‘𝑥))))
10		elun 4176	. . . . . . . . 9 ⊢ (𝐴 ∈ (( O ‘( bday ‘𝑥)) ∪ ( N ‘( bday ‘𝑥))) ↔ (𝐴 ∈ ( O ‘( bday ‘𝑥)) ∨ 𝐴 ∈ ( N ‘( bday ‘𝑥))))
11	9, 10	bitri 275	. . . . . . . 8 ⊢ (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ (𝐴 ∈ ( O ‘( bday ‘𝑥)) ∨ 𝐴 ∈ ( N ‘( bday ‘𝑥))))
12		lrold 27953	. . . . . . . . . . 11 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday ‘𝑥))
13	12	eleq2i 2836	. . . . . . . . . 10 ⊢ (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ↔ 𝐴 ∈ ( O ‘( bday ‘𝑥)))
14		elons 28294	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Ons ↔ (𝑥 ∈ No ∧ ( R ‘𝑥) = ∅))
15	14	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Ons → ( R ‘𝑥) = ∅)
16	15	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ( R ‘𝑥) = ∅)
17	16	uneq2d 4191	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( L ‘𝑥) ∪ ∅))
18		un0 4417	. . . . . . . . . . . . 13 ⊢ (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
19	17, 18	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( L ‘𝑥))
20	19	eleq2d 2830	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ↔ 𝐴 ∈ ( L ‘𝑥)))
21		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝐴 ∈ No )
22		onsno 28296	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Ons → 𝑥 ∈ No )
23	22	ad2antlr 726	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝑥 ∈ No )
24		breq1 5169	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 = 𝐴 → (𝑥𝑂 <s 𝑥 ↔ 𝐴 <s 𝑥))
25		leftval 27920	. . . . . . . . . . . . . . . 16 ⊢ ( L ‘𝑥) = {𝑥𝑂 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥𝑂 <s 𝑥}
26	24, 25	elrab2 3711	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ( L ‘𝑥) ↔ (𝐴 ∈ ( O ‘( bday ‘𝑥)) ∧ 𝐴 <s 𝑥))
27	26	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ( L ‘𝑥) → 𝐴 <s 𝑥)
28	27	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝐴 <s 𝑥)
29	21, 23, 28	sltled 27832	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝐴 ≤s 𝑥)
30	29	ex 412	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( L ‘𝑥) → 𝐴 ≤s 𝑥))
31	20, 30	sylbid 240	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) → 𝐴 ≤s 𝑥))
32	13, 31	biimtrrid 243	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( O ‘( bday ‘𝑥)) → 𝐴 ≤s 𝑥))
33		newbday 27958	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝐴 ∈ No ) → (𝐴 ∈ ( N ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) = ( bday ‘𝑥)))
34	1, 33	mpan 689	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → (𝐴 ∈ ( N ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) = ( bday ‘𝑥)))
35	34	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( N ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) = ( bday ‘𝑥)))
36		leftssold 27935	. . . . . . . . . . . . 13 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
37		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝐴) = ( bday ‘𝑥) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝑥)))
38	37	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝑥)))
39	15	uneq2d 4191	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Ons → (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( L ‘𝑥) ∪ ∅))
40	39, 12, 18	3eqtr3g 2803	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Ons → ( O ‘( bday ‘𝑥)) = ( L ‘𝑥))
41	40	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( O ‘( bday ‘𝑥)) = ( L ‘𝑥))
42	38, 41	eqtr2d 2781	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( L ‘𝑥) = ( O ‘( bday ‘𝐴)))
43	36, 42	sseqtrrid 4062	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( L ‘𝐴) ⊆ ( L ‘𝑥))
44		slelss 27964	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
45	22, 44	syl3an2 1164	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
46	45	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
47	43, 46	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → 𝐴 ≤s 𝑥)
48	47	ex 412	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( bday ‘𝐴) = ( bday ‘𝑥) → 𝐴 ≤s 𝑥))
49	35, 48	sylbid 240	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( N ‘( bday ‘𝑥)) → 𝐴 ≤s 𝑥))
50	32, 49	jaod 858	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((𝐴 ∈ ( O ‘( bday ‘𝑥)) ∨ 𝐴 ∈ ( N ‘( bday ‘𝑥))) → 𝐴 ≤s 𝑥))
51	11, 50	biimtrid 242	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( M ‘( bday ‘𝑥)) → 𝐴 ≤s 𝑥))
52		madebday 27956	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝐴 ∈ No ) → (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
53	1, 52	mpan 689	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
54	53	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
55		slenlt 27815	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ) → (𝐴 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝐴))
56	22, 55	sylan2 592	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝐴))
57	51, 54, 56	3imtr3d 293	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( bday ‘𝐴) ⊆ ( bday ‘𝑥) → ¬ 𝑥 <s 𝐴))
58	57	con2d 134	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝑥 <s 𝐴 → ¬ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
59	58	3impia 1117	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → ¬ ( bday ‘𝐴) ⊆ ( bday ‘𝑥))
60	7, 59	olcnd 876	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
61	22	3ad2ant2 1134	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → 𝑥 ∈ No )
62		oldbday 27957	. . . 4 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐴)))
63	3, 61, 62	sylancr 586	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → (𝑥 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐴)))
64	60, 63	mpbird 257	. 2 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → 𝑥 ∈ ( O ‘( bday ‘𝐴)))
65	64	rabssdv 4098	1 ⊢ (𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ ( O ‘( bday ‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {crab 3443   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166  Ord word 6394  Oncon0 6395  ‘cfv 6573   No csur 27702   <s cslt 27703   bday cbday 27704   ≤s csle 27807   M cmade 27899   O cold 27900   N cnew 27901   L cleft 27902   R cright 27903  On_scons 28292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-made 27904  df-old 27905  df-new 27906  df-left 27907  df-right 27908  df-ons 28293

This theorem is referenced by:  sltonex  28302