Theorem bdayfin 28501

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 28499	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)
2		bdayfinlem 28500	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])
3	2	3exp 1126	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 → (( bday ‘𝐴) ∈ ω → 𝐴 ∈ ℤs[1/2])))
4		negscl 28050	. . . . . 6 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
5		bdayfinlem 28500	. . . . . . 7 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s ≤s ( -us ‘𝐴) ∧ ( bday ‘( -us ‘𝐴)) ∈ ω) → ( -us ‘𝐴) ∈ ℤs[1/2])
6	5	3expib 1129	. . . . . 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 27823	. . . . . . . 8 ⊢ 0s ∈ No
9		lenegs 28060	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
10	8, 9	mpan2 698	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
11		neg0s 28040	. . . . . . . 8 ⊢ ( -us ‘ 0s ) = 0s
12	11	breq1i 5082	. . . . . . 7 ⊢ (( -us ‘ 0s ) ≤s ( -us ‘𝐴) ↔ 0s ≤s ( -us ‘𝐴))
13	10, 12	bitrdi 289	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ 0s ≤s ( -us ‘𝐴)))
14		negbday 28071	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
15	14	eqcomd 2747	. . . . . . 7 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ( bday ‘( -us ‘𝐴)))
16	15	eleq1d 2826	. . . . . 6 ⊢ (𝐴 ∈ No → (( bday ‘𝐴) ∈ ω ↔ ( bday ‘( -us ‘𝐴)) ∈ ω))
17	13, 16	anbi12d 639	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴 ≤s 0s ∧ ( bday ‘𝐴) ∈ ω) ↔ ( 0s ≤s ( -us ‘𝐴) ∧ ( bday ‘( -us ‘𝐴)) ∈ ω)))
18		z12negsclb 28495	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ∈ ℤs[1/2] ↔ ( -us ‘𝐴) ∈ ℤs[1/2]))
19	7, 17, 18	3imtr4d 296	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ≤s 0s ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2]))
20	19	expd 417	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s → (( bday ‘𝐴) ∈ ω → 𝐴 ∈ ℤs[1/2])))
21		lestric 27754	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
22	8, 21	mpan 697	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
23	3, 20, 22	mpjaod 867	. 2 ⊢ (𝐴 ∈ No → (( bday ‘𝐴) ∈ ω → 𝐴 ∈ ℤs[1/2]))
24	1, 23	impbid2 228	1 ⊢ (𝐴 ∈ No → (𝐴 ∈ ℤs[1/2] ↔ ( bday ‘𝐴) ∈ ω))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   ∈ wcel 2121   class class class wbr 5075  ‘cfv 6489  ωcom 7810   No csur 27625   bday cbday 27627   ≤s cles 27730   0_s c0s 27819   -u_s cnegs 28033  ℤ_s[1/2]cz12s 28428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-dc 10363  ax-ac2 10380

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-fin 8891  df-card 9858  df-acn 9861  df-ac 10033  df-no 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-0s 27821  df-1s 27822  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-norec 27952  df-norec2 27963  df-adds 27974  df-negs 28035  df-subs 28036  df-muls 28121  df-divs 28202  df-ons 28266  df-seqs 28298  df-n0s 28328  df-nns 28329  df-zs 28393  df-2s 28425  df-exps 28427  df-z12s 28429

This theorem is referenced by:  dfz12s2  28502