Theorem onsbnd2 28292

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 27853	. . . . . . 7 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
2	1	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
3		negbday 28067	. . . . . 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 27898	. . . . . 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 3957	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘( -us ‘𝐵)) ⊆ ( bday ‘𝐴))
8		bdayon 27762	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9	2	negscld 28047	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐵) ∈ No )
10		madebday 27910	. . . . 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 28291	. . 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 28265	. . . . . 6 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
16	15	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 ∈ No )
17	16	negscld 28047	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ∈ No )
18	17, 2	lenegsd 28058	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( -us ‘𝐴) ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘( -us ‘𝐴))))
19		negnegs 28054	. . . . 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 5098	. . 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 3890   class class class wbr 5086  Oncon0 6319  ‘cfv 6494   No csur 27621   bday cbday 27623   ≤s cles 27726   M cmade 27832   -u_s cnegs 28029  On_scons 28261

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-1o 8400  df-2o 8401  df-nadd 8597  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-ons 28262

This theorem is referenced by: (None)