Theorem onsbnd2 28443

Description: The surreals of a given birthday are bounded below by the negative of that ordinal. (Contributed by Scott Fenton, 22-Feb-2026.)

Assertion

Ref	Expression
onsbnd2	⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ≤s 𝐵)

Proof of Theorem onsbnd2

Step	Hyp	Ref	Expression
1		madeno 28004	. . . . . . 7 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
2	1	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
3		negbday 28218	. . . . . 6 ⊢ (𝐵 ∈ No → ( bday ‘( -us ‘𝐵)) = ( bday ‘𝐵))
4	2, 3	syl 18	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘( -us ‘𝐵)) = ( bday ‘𝐵))
5		madebdayim 28049	. . . . . 6 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
6	5	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
7	4, 6	eqsstrd 3979	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴))
8		bdayon 27913	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9	2	negscld 28198	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ∈ No )
10		madebday 28061	. . . . 5 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝐵) ∈ No ) → (( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴)))
11	8, 9, 10	sylancr 598	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴)))
12	7, 11	mpbird 260	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴)))
13		onsbnd 28442	. . 3 ⊢ ((𝐴 ∈ Ons ∧ ( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ≤s 𝐴)
14	12, 13	syldan 602	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ≤s 𝐴)
15		onno 28416	. . . . . 6 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
16	15	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 ∈ No )
17	16	negscld 28198	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ∈ No )
18	17, 2	lenegsd 28209	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐴) ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘( -us ‘𝐴))))
19		negnegs 28205	. . . . 5 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
20	16, 19	syl 18	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
21	20	breq2d 5125	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐵) ≤s ( -us ‘( -us ‘𝐴)) ↔ ( -us ‘𝐵) ≤s 𝐴))
22	18, 21	bitr2d 283	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐵) ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s 𝐵))
23	14, 22	mpbid 235	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ≤s 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913   class class class wbr 5113  Oncon0 6363  ‘cfv 6539   No csur 27772   bday cbday 27774   ≤s cles 27876   M cmade 27983   -u_s cnegs 28180  On_scons 28412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27775  df-lts 27776  df-bday 27777  df-les 27877  df-slts 27919  df-cuts 27921  df-0s 27968  df-made 27988  df-old 27989  df-left 27991  df-right 27992  df-norec 28099  df-norec2 28110  df-adds 28121  df-negs 28182  df-ons 28413

This theorem is referenced by: (None)