Theorem onsbnd2 28543

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 28104	. . . . . . 7 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
2	1	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
3		negbday 28318	. . . . . 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 28149	. . . . . 6 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
6	5	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
7	4, 6	eqsstrd 3968	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴))
8		bdayon 28013	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9	2	negscld 28298	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ∈ No )
10		madebday 28161	. . . . 5 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝐵) ∈ No ) → (( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴)))
11	8, 9, 10	sylancr 599	. . . 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 28542	. . 3 ⊢ ((𝐴 ∈ Ons ∧ ( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ≤s 𝐴)
14	12, 13	syldan 603	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ≤s 𝐴)
15		onno 28516	. . . . . 6 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
16	15	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 ∈ No )
17	16	negscld 28298	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ∈ No )
18	17, 2	lenegsd 28309	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐴) ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘( -us ‘𝐴))))
19		negnegs 28305	. . . . 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 5119	. . 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 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902   class class class wbr 5107  Oncon0 6361  ‘cfv 6537   No csur 27872   bday cbday 27874   ≤s cles 27976   M cmade 28083   -u_s cnegs 28280  On_scons 28512

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8458  df-2o 8459  df-nadd 8657  df-no 27875  df-lts 27876  df-bday 27877  df-les 27977  df-slts 28019  df-cuts 28021  df-0s 28068  df-made 28088  df-old 28089  df-left 28091  df-right 28092  df-norec 28199  df-norec2 28210  df-adds 28221  df-negs 28282  df-ons 28513

This theorem is used by: (None)