Theorem onsbnd2 28484

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 28045	. . . . . . 7 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
2	1	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
3		negbday 28259	. . . . . 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 28090	. . . . . 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 3971	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴))
8		bdayon 27954	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9	2	negscld 28239	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ∈ No )
10		madebday 28102	. . . . 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 28483	. . 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 28457	. . . . . 6 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
16	15	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 ∈ No )
17	16	negscld 28239	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ∈ No )
18	17, 2	lenegsd 28250	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐴) ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘( -us ‘𝐴))))
19		negnegs 28246	. . . . 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 5121	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905   class class class wbr 5109  Oncon0 6360  ‘cfv 6536   No csur 27813   bday cbday 27815   ≤s cles 27917   M cmade 28024   -u_s cnegs 28221  On_scons 28453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-1o 8449  df-2o 8450  df-nadd 8648  df-no 27816  df-lts 27817  df-bday 27818  df-les 27918  df-slts 27960  df-cuts 27962  df-0s 28009  df-made 28029  df-old 28030  df-left 28032  df-right 28033  df-norec 28140  df-norec2 28151  df-adds 28162  df-negs 28223  df-ons 28454

This theorem is used by: (None)