Theorem sltsbday 27909

Description: Birthday comparison rule for surreals. (Contributed by Scott Fenton, 23-Feb-2026.)

Hypotheses

Ref	Expression
sltsbday.1	⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
sltsbday.2	⊢ (𝜑 → 𝐵 ∈ No )
sltsbday.3	⊢ (𝜑 → 𝐿 <<s {𝐵})
sltsbday.4	⊢ (𝜑 → {𝐵} <<s 𝑅)

Assertion

Ref	Expression
sltsbday	⊢ (𝜑 → ( bday ‘𝐴) ⊆ ( bday ‘𝐵))

Proof of Theorem sltsbday

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sltsbday.1	. . 3 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
2	1	fveq2d 6844	. 2 ⊢ (𝜑 → ( bday ‘𝐴) = ( bday ‘(𝐿 |s 𝑅)))
3		sltsbday.3	. . . . 5 ⊢ (𝜑 → 𝐿 <<s {𝐵})
4		sltsbday.4	. . . . 5 ⊢ (𝜑 → {𝐵} <<s 𝑅)
5		sltsbday.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
6	5	snn0d 4719	. . . . 5 ⊢ (𝜑 → {𝐵} ≠ ∅)
7		sltstr 27779	. . . . 5 ⊢ ((𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅 ∧ {𝐵} ≠ ∅) → 𝐿 <<s 𝑅)
8	3, 4, 6, 7	syl3anc 1374	. . . 4 ⊢ (𝜑 → 𝐿 <<s 𝑅)
9		cutbday 27776	. . . 4 ⊢ (𝐿 <<s 𝑅 → ( bday ‘(𝐿 |s 𝑅)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → ( bday ‘(𝐿 |s 𝑅)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
11		bdayfn 27741	. . . . 5 ⊢ bday Fn No
12		ssrab2 4020	. . . . 5 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
13		sneq 4577	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
14	13	breq2d 5097	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝐵}))
15	13	breq1d 5095	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ({𝑥} <<s 𝑅 ↔ {𝐵} <<s 𝑅))
16	14, 15	anbi12d 633	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅)))
17	3, 4	jca 511	. . . . . 6 ⊢ (𝜑 → (𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅))
18	16, 5, 17	elrabd 3636	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
19		fnfvima 7188	. . . . 5 ⊢ (( bday Fn No ∧ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No ∧ 𝐵 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ( bday ‘𝐵) ∈ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
20	11, 12, 18, 19	mp3an12i 1468	. . . 4 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
21		intss1 4905	. . . 4 ⊢ (( bday ‘𝐵) ∈ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ∩ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ ( bday ‘𝐵))
22	20, 21	syl 17	. . 3 ⊢ (𝜑 → ∩ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ ( bday ‘𝐵))
23	10, 22	eqsstrd 3956	. 2 ⊢ (𝜑 → ( bday ‘(𝐿 |s 𝑅)) ⊆ ( bday ‘𝐵))
24	2, 23	eqsstrd 3956	1 ⊢ (𝜑 → ( bday ‘𝐴) ⊆ ( bday ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  {crab 3389   ⊆ wss 3889  ∅c0 4273  {csn 4567  ∩ cint 4889   class class class wbr 5085   “ cima 5634   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   No csur 27603   bday cbday 27605   <<s cslts 27749   |s ccuts 27751

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

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-uni 4851  df-int 4890  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-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-ord 6326  df-on 6327  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-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752

This theorem is referenced by:  bdayfinbndlem1  28459