Theorem bdayfin 28500

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 28498	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)
2		bdayfinlem 28499	. . . 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 28049	. . . . . 6 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
5		bdayfinlem 28499	. . . . . . 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 27822	. . . . . . . 8 ⊢ 0s ∈ No
9		lenegs 28059	. . . . . . . 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 28039	. . . . . . . 8 ⊢ ( -us ‘ 0s ) = 0s
12	11	breq1i 5107	. . . . . . 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 28070	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
15	14	eqcomd 2743	. . . . . . 7 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ( bday ‘( -us ‘𝐴)))
16	15	eleq1d 2822	. . . . . 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 28494	. . . . 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 27753	. . . 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 5100  ‘cfv 6502  ωcom 7820   No csur 27624   bday cbday 27626   ≤s cles 27729   0_s c0s 27818   -u_s cnegs 28032  ℤ_s[1/2]cz12s 28427

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-dc 10370  ax-ac2 10387

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-oadd 8413  df-nadd 8606  df-er 8647  df-map 8779  df-en 8898  df-dom 8899  df-fin 8901  df-card 9865  df-acn 9868  df-ac 10040  df-no 27627  df-lts 27628  df-bday 27629  df-les 27730  df-slts 27771  df-cuts 27773  df-0s 27820  df-1s 27821  df-made 27840  df-old 27841  df-left 27843  df-right 27844  df-norec 27951  df-norec2 27962  df-adds 27973  df-negs 28034  df-subs 28035  df-muls 28120  df-divs 28201  df-ons 28265  df-seqs 28297  df-n0s 28327  df-nns 28328  df-zs 28392  df-2s 28424  df-exps 28426  df-z12s 28428

This theorem is referenced by:  dfz12s2  28501