Theorem bdayfin 28464

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		zs12bday 28462	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)
2		bdayfinlem 28463	. . . 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 28016	. . . . . 6 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
5		bdayfinlem 28463	. . . . . . 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		0sno 27805	. . . . . . . 8 ⊢ 0s ∈ No
9		sleneg 28026	. . . . . . . 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		negs0s 28006	. . . . . . . 8 ⊢ ( -us ‘ 0s ) = 0s
12	11	breq1i 5104	. . . . . . 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		negsbday 28037	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
15	14	eqcomd 2741	. . . . . . 7 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ( bday ‘( -us ‘𝐴)))
16	15	eleq1d 2820	. . . . . 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		zs12negsclb 28458	. . . . 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		sletric 27738	. . . 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 5097  ‘cfv 6491  ωcom 7808   No csur 27609   bday cbday 27611   ≤s csle 27714   0_s c0s 27801   -u_s cnegs 27999  ℤ_s[1/2]czs12 28391

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-dc 10358  ax-ac2 10375

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  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 8767  df-en 8886  df-dom 8887  df-fin 8889  df-card 9853  df-acn 9856  df-ac 10028  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27918  df-norec2 27929  df-adds 27940  df-negs 28001  df-subs 28002  df-muls 28087  df-divs 28168  df-ons 28231  df-seqs 28263  df-n0s 28293  df-nns 28294  df-zs 28356  df-2s 28388  df-exps 28390  df-zs12 28392

This theorem is referenced by:  dfzs122  28465