Theorem sltsbday 28068

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 6875	. 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 4737	. . . . 5 ⊢ (𝜑 → {𝐵} ≠ ∅)
7		sltstr 27938	. . . . 5 ⊢ ((𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅 ∧ {𝐵} ≠ ∅) → 𝐿 <<s 𝑅)
8	3, 4, 6, 7	syl3anc 1394	. . . 4 ⊢ (𝜑 → 𝐿 <<s 𝑅)
9		cutbday 27935	. . . 4 ⊢ (𝐿 <<s 𝑅 → ( bday ‘(𝐿 |s 𝑅)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
10	8, 9	syl 18	. . 3 ⊢ (𝜑 → ( bday ‘(𝐿 |s 𝑅)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
11		bdayfn 27899	. . . . 5 ⊢ bday Fn No
12		ssrab2 4036	. . . . 5 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
13		sneq 4595	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
14	13	breq2d 5117	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝐵}))
15	13	breq1d 5115	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ({𝑥} <<s 𝑅 ↔ {𝐵} <<s 𝑅))
16	14, 15	anbi12d 643	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅)))
17	3, 4	jca 520	. . . . . 6 ⊢ (𝜑 → (𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅))
18	16, 5, 17	elrabd 3655	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
19		fnfvima 7221	. . . . 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 1489	. . . 4 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
21		intss1 4924	. . . 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 18	. . 3 ⊢ (𝜑 → ∩ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ ( bday ‘𝐵))
23	10, 22	eqsstrd 3973	. 2 ⊢ (𝜑 → ( bday ‘(𝐿 |s 𝑅)) ⊆ ( bday ‘𝐵))
24	2, 23	eqsstrd 3973	1 ⊢ (𝜑 → ( bday ‘𝐴) ⊆ ( bday ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  {crab 3417   ⊆ wss 3907  ∅c0 4288  {csn 4585  ∩ cint 4908   class class class wbr 5105   “ cima 5655   Fn wfn 6520  ‘cfv 6525  (class class class)co 7400   No csur 27762   bday cbday 27764   <<s cslts 27908   |s ccuts 27910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1o 8441  df-2o 8442  df-no 27765  df-lts 27766  df-bday 27767  df-slts 27909  df-cuts 27911

This theorem is referenced by:  bdayfinbndlem1  28618