Theorem ltonold 28257

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 27748	. . . . . . 7 ⊢ ( bday ‘𝑥) ∈ On
2	1	onordi 6430	. . . . . 6 ⊢ Ord ( bday ‘𝑥)
3		bdayon 27748	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
4	3	onordi 6430	. . . . . 6 ⊢ Ord ( bday ‘𝐴)
5		ordtri2or 6417	. . . . . 6 ⊢ ((Ord ( bday ‘𝑥) ∧ Ord ( bday ‘𝐴)) → (( bday ‘𝑥) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
6	2, 4, 5	mp2an 692	. . . . 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 27880	. . . . . . . . . 10 ⊢ ( M ‘( bday ‘𝑥)) = (( O ‘( bday ‘𝑥)) ∪ ( N ‘( bday ‘𝑥)))
9	8	eleq2i 2828	. . . . . . . . 9 ⊢ (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝐴 ∈ (( O ‘( bday ‘𝑥)) ∪ ( N ‘( bday ‘𝑥))))
10		elun 4105	. . . . . . . . 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 27893	. . . . . . . . . . 11 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday ‘𝑥))
13	12	eleq2i 2828	. . . . . . . . . 10 ⊢ (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ↔ 𝐴 ∈ ( O ‘( bday ‘𝑥)))
14		elons 28249	. . . . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( L ‘𝑥) ∪ ∅))
18		un0 4346	. . . . . . . . . . . . 13 ⊢ (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
19	17, 18	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( L ‘𝑥))
20	19	eleq2d 2822	. . . . . . . . . . 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		onno 28251	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Ons → 𝑥 ∈ No )
23	22	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝑥 ∈ No )
24		leftlt 27849	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ( L ‘𝑥) → 𝐴 <s 𝑥)
25	24	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝐴 <s 𝑥)
26	21, 23, 25	ltlesd 27741	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝐴 ≤s 𝑥)
27	26	ex 412	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( L ‘𝑥) → 𝐴 ≤s 𝑥))
28	20, 27	sylbid 240	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) → 𝐴 ≤s 𝑥))
29	13, 28	biimtrrid 243	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( O ‘( bday ‘𝑥)) → 𝐴 ≤s 𝑥))
30		newbday 27898	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝐴 ∈ No ) → (𝐴 ∈ ( N ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) = ( bday ‘𝑥)))
31	1, 30	mpan 690	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → (𝐴 ∈ ( N ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) = ( bday ‘𝑥)))
32	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( N ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) = ( bday ‘𝑥)))
33		leftssold 27867	. . . . . . . . . . . . 13 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
34		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝐴) = ( bday ‘𝑥) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝑥)))
35	34	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝑥)))
36		onleft 28256	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Ons → ( O ‘( bday ‘𝑥)) = ( L ‘𝑥))
37	36	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( O ‘( bday ‘𝑥)) = ( L ‘𝑥))
38	35, 37	eqtr2d 2772	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( L ‘𝑥) = ( O ‘( bday ‘𝐴)))
39	33, 38	sseqtrrid 3977	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( L ‘𝐴) ⊆ ( L ‘𝑥))
40		leslss 27905	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
41	22, 40	syl3an2 1164	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
42	41	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
43	39, 42	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → 𝐴 ≤s 𝑥)
44	43	ex 412	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( bday ‘𝐴) = ( bday ‘𝑥) → 𝐴 ≤s 𝑥))
45	32, 44	sylbid 240	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( N ‘( bday ‘𝑥)) → 𝐴 ≤s 𝑥))
46	29, 45	jaod 859	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → ((𝐴 ∈ ( O ‘( bday ‘𝑥)) ∨ 𝐴 ∈ ( N ‘( bday ‘𝑥))) → 𝐴 ≤s 𝑥))
47	11, 46	biimtrid 242	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( M ‘( bday ‘𝑥)) → 𝐴 ≤s 𝑥))
48		madebday 27896	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∈ On ∧ 𝐴 ∈ No ) → (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
49	1, 48	mpan 690	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
50	49	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
51		lenlts 27720	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ No ) → (𝐴 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝐴))
52	22, 51	sylan2 593	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝐴 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝐴))
53	47, 50, 52	3imtr3d 293	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( bday ‘𝐴) ⊆ ( bday ‘𝑥) → ¬ 𝑥 <s 𝐴))
54	53	con2d 134	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (𝑥 <s 𝐴 → ¬ ( bday ‘𝐴) ⊆ ( bday ‘𝑥)))
55	54	3impia 1117	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → ¬ ( bday ‘𝐴) ⊆ ( bday ‘𝑥))
56	7, 55	olcnd 877	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
57	22	3ad2ant2 1134	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → 𝑥 ∈ No )
58		oldbday 27897	. . . 4 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐴)))
59	3, 57, 58	sylancr 587	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → (𝑥 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐴)))
60	56, 59	mpbird 257	. 2 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝐴) → 𝑥 ∈ ( O ‘( bday ‘𝐴)))
61	60	rabssdv 4026	1 ⊢ (𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ ( O ‘( bday ‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {crab 3399   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285   class class class wbr 5098  Ord word 6316  Oncon0 6317  ‘cfv 6492   No csur 27607   <s clts 27608   bday cbday 27609   ≤s cles 27712   M cmade 27818   O cold 27819   N cnew 27820   L cleft 27821   R cright 27822  On_scons 28247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-made 27823  df-old 27824  df-new 27825  df-left 27826  df-right 27827  df-ons 28248

This theorem is referenced by:  ltonsex  28258  onsfi  28352