Theorem bdayfin 28483

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 28481	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)
2		bdayfinlem 28482	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])
3	2	3exp 1119	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 → (( bday ‘𝐴) ∈ ω → 𝐴 ∈ ℤs[1/2])))
4		negscl 28032	. . . . . 6 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
5		bdayfinlem 28482	. . . . . . 7 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s ≤s ( -us ‘𝐴) ∧ ( bday ‘( -us ‘𝐴)) ∈ ω) → ( -us ‘𝐴) ∈ ℤs[1/2])
6	5	3expib 1122	. . . . . 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 27805	. . . . . . . 8 ⊢ 0s ∈ No
9		lenegs 28042	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
10	8, 9	mpan2 691	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
11		neg0s 28022	. . . . . . . 8 ⊢ ( -us ‘ 0s ) = 0s
12	11	breq1i 5105	. . . . . . 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 28053	. . . . . . . 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 632	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴 ≤s 0s ∧ ( bday ‘𝐴) ∈ ω) ↔ ( 0s ≤s ( -us ‘𝐴) ∧ ( bday ‘( -us ‘𝐴)) ∈ ω)))
18		z12negsclb 28477	. . . . 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 27736	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
22	8, 21	mpan 690	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
23	3, 20, 22	mpjaod 860	. 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 847   ∈ wcel 2113   class class class wbr 5098  ‘cfv 6492  ωcom 7808   No csur 27607   bday cbday 27609   ≤s cles 27712   0_s c0s 27801   -u_s cnegs 28015  ℤ_s[1/2]cz12s 28410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-dc 10356  ax-ac2 10373

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-fin 8887  df-card 9851  df-acn 9854  df-ac 10026  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27934  df-norec2 27945  df-adds 27956  df-negs 28017  df-subs 28018  df-muls 28103  df-divs 28184  df-ons 28248  df-seqs 28280  df-n0s 28310  df-nns 28311  df-zs 28375  df-2s 28407  df-exps 28409  df-z12s 28411

This theorem is referenced by:  dfz12s2  28484