Theorem sltonold 28198

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

Step	Hyp	Ref	Expression
1		bdayelon 27715	. . . . . . 7 ⊢ ( bday ‘𝑥) ∈ On
2	1	onordi 6419	. . . . . 6 ⊢ Ord ( bday ‘𝑥)
3		bdayelon 27715	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
4	3	onordi 6419	. . . . . 6 ⊢ Ord ( bday ‘𝐴)
5		ordtri2or 6406	. . . . . 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 27829	. . . . . . . . . 10 ⊢ ( M ‘( bday ‘𝑥)) = (( O ‘( bday ‘𝑥)) ∪ ( N ‘( bday ‘𝑥)))
9	8	eleq2i 2823	. . . . . . . . 9 ⊢ (𝐴 ∈ ( M ‘( bday ‘𝑥)) ↔ 𝐴 ∈ (( O ‘( bday ‘𝑥)) ∪ ( N ‘( bday ‘𝑥))))
10		elun 4100	. . . . . . . . 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 27842	. . . . . . . . . . 11 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday ‘𝑥))
13	12	eleq2i 2823	. . . . . . . . . 10 ⊢ (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ↔ 𝐴 ∈ ( O ‘( bday ‘𝑥)))
14		elons 28190	. . . . . . . . . . . . . . . 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 4115	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( L ‘𝑥) ∪ ∅))
18		un0 4341	. . . . . . . . . . . . 13 ⊢ (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
19	17, 18	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( L ‘𝑥))
20	19	eleq2d 2817	. . . . . . . . . . 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 28192	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Ons → 𝑥 ∈ No )
23	22	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝑥 ∈ No )
24		leftlt 27808	. . . . . . . . . . . . . 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	sltled 27708	. . . . . . . . . . . 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 27847	. . . . . . . . . . . 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 27824	. . . . . . . . . . . . 13 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
34		fveq2 6822	. . . . . . . . . . . . . . 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		onsleft 28197	. . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( L ‘𝑥) = ( O ‘( bday ‘𝐴)))
39	33, 38	sseqtrrid 3973	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥 ∈ Ons) ∧ ( bday ‘𝐴) = ( bday ‘𝑥)) → ( L ‘𝐴) ⊆ ( L ‘𝑥))
40		slelss 27854	. . . . . . . . . . . . . 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 27845	. . . . . . . . 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		slenlt 27691	. . . . . . . 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 27846	. . . 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 4020	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 2111  {crab 3395   ∪ cun 3895   ⊆ wss 3897  ∅c0 4280   class class class wbr 5089  Ord word 6305  Oncon0 6306  ‘cfv 6481   No csur 27578   <s cslt 27579   bday cbday 27580   ≤s csle 27683   M cmade 27783   O cold 27784   N cnew 27785   L cleft 27786   R cright 27787  On_scons 28188

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-made 27788  df-old 27789  df-new 27790  df-left 27791  df-right 27792  df-ons 28189

This theorem is referenced by:  sltonex  28199  onsfi  28283