Theorem onsbnd2 28274

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 27835	. . . . . . 7 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
2	1	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
3		negbday 28049	. . . . . 6 ⊢ (𝐵 ∈ No → ( bday ‘( -us ‘𝐵)) = ( bday ‘𝐵))
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘( -us ‘𝐵)) = ( bday ‘𝐵))
5		madebdayim 27880	. . . . . 6 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
6	5	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
7	4, 6	eqsstrd 3956	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴))
8		bdayon 27744	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9	2	negscld 28029	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ∈ No )
10		madebday 27892	. . . . 5 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝐵) ∈ No ) → (( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴)))
11	8, 9, 10	sylancr 588	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴)))
12	7, 11	mpbird 257	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴)))
13		onsbnd 28273	. . 3 ⊢ ((𝐴 ∈ Ons ∧ ( -us ‘𝐵) ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ≤s 𝐴)
14	12, 13	syldan 592	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ≤s 𝐴)
15		onno 28247	. . . . . 6 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
16	15	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 ∈ No )
17	16	negscld 28029	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ∈ No )
18	17, 2	lenegsd 28040	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐴) ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘( -us ‘𝐴))))
19		negnegs 28036	. . . . 5 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
20	16, 19	syl 17	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
21	20	breq2d 5097	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐵) ≤s ( -us ‘( -us ‘𝐴)) ↔ ( -us ‘𝐵) ≤s 𝐴))
22	18, 21	bitr2d 280	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐵) ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s 𝐵))
23	14, 22	mpbid 232	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ≤s 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889   class class class wbr 5085  Oncon0 6323  ‘cfv 6498   No csur 27603   bday cbday 27605   ≤s cles 27708   M cmade 27814   -u_s cnegs 28011  On_scons 28243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-ons 28244

This theorem is referenced by: (None)