Theorem bdayfin 28479

Description: A surreal has a finite birthday iff it is a dyadic fraction. (Contributed by Scott Fenton, 26-Feb-2026.)

Assertion

Ref	Expression
bdayfin	⊢ (𝐴 ∈ No → (𝐴 ∈ ℤs[1/2] ↔ ( bday ‘𝐴) ∈ ω))

Proof of Theorem bdayfin

Step	Hyp	Ref	Expression
1		z12bday 28477	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)
2		bdayfinlem 28478	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])
3	2	3exp 1120	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 → (( bday ‘𝐴) ∈ ω → 𝐴 ∈ ℤs[1/2])))
4		negscl 28028	. . . . . 6 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
5		bdayfinlem 28478	. . . . . . 7 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s ≤s ( -us ‘𝐴) ∧ ( bday ‘( -us ‘𝐴)) ∈ ω) → ( -us ‘𝐴) ∈ ℤs[1/2])
6	5	3expib 1123	. . . . . 6 ⊢ (( -us ‘𝐴) ∈ No → (( 0s ≤s ( -us ‘𝐴) ∧ ( bday ‘( -us ‘𝐴)) ∈ ω) → ( -us ‘𝐴) ∈ ℤs[1/2]))
7	4, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → (( 0s ≤s ( -us ‘𝐴) ∧ ( bday ‘( -us ‘𝐴)) ∈ ω) → ( -us ‘𝐴) ∈ ℤs[1/2]))
8		0no 27801	. . . . . . . 8 ⊢ 0s ∈ No
9		lenegs 28038	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
10	8, 9	mpan2 692	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
11		neg0s 28018	. . . . . . . 8 ⊢ ( -us ‘ 0s ) = 0s
12	11	breq1i 5092	. . . . . . 7 ⊢ (( -us ‘ 0s ) ≤s ( -us ‘𝐴) ↔ 0s ≤s ( -us ‘𝐴))
13	10, 12	bitrdi 287	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ 0s ≤s ( -us ‘𝐴)))
14		negbday 28049	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
15	14	eqcomd 2742	. . . . . . 7 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ( bday ‘( -us ‘𝐴)))
16	15	eleq1d 2821	. . . . . 6 ⊢ (𝐴 ∈ No → (( bday ‘𝐴) ∈ ω ↔ ( bday ‘( -us ‘𝐴)) ∈ ω))
17	13, 16	anbi12d 633	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴 ≤s 0s ∧ ( bday ‘𝐴) ∈ ω) ↔ ( 0s ≤s ( -us ‘𝐴) ∧ ( bday ‘( -us ‘𝐴)) ∈ ω)))
18		z12negsclb 28473	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ∈ ℤs[1/2] ↔ ( -us ‘𝐴) ∈ ℤs[1/2]))
19	7, 17, 18	3imtr4d 294	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ≤s 0s ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2]))
20	19	expd 415	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s → (( bday ‘𝐴) ∈ ω → 𝐴 ∈ ℤs[1/2])))
21		lestric 27732	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
22	8, 21	mpan 691	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
23	3, 20, 22	mpjaod 861	. 2 ⊢ (𝐴 ∈ No → (( bday ‘𝐴) ∈ ω → 𝐴 ∈ ℤs[1/2]))
24	1, 23	impbid2 226	1 ⊢ (𝐴 ∈ No → (𝐴 ∈ ℤs[1/2] ↔ ( bday ‘𝐴) ∈ ω))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∈ wcel 2114   class class class wbr 5085  ‘cfv 6498  ωcom 7817   No csur 27603   bday cbday 27605   ≤s cles 27708   0_s c0s 27797   -u_s cnegs 28011  ℤ_s[1/2]cz12s 28406

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  ax-dc 10368  ax-ac2 10385

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-lim 6328  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-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-nadd 8602  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-fin 8897  df-card 9863  df-acn 9866  df-ac 10038  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  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-subs 28014  df-muls 28099  df-divs 28180  df-ons 28244  df-seqs 28276  df-n0s 28306  df-nns 28307  df-zs 28371  df-2s 28403  df-exps 28405  df-z12s 28407

This theorem is referenced by:  dfz12s2  28480