Theorem sltsbday 28190

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 6886	. 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 4739	. . . . 5 ⊢ (𝜑 → {𝐵} ≠ ∅)
7		sltstr 28060	. . . . 5 ⊢ ((𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅 ∧ {𝐵} ≠ ∅) → 𝐿 <<s 𝑅)
8	3, 4, 6, 7	syl3anc 1398	. . . 4 ⊢ (𝜑 → 𝐿 <<s 𝑅)
9		cutbday 28057	. . . 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 28021	. . . . 5 ⊢ bday Fn No
12		ssrab2 4031	. . . . 5 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
13		sneq 4597	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
14	13	breq2d 5119	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝐵}))
15	13	breq1d 5117	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ({𝑥} <<s 𝑅 ↔ {𝐵} <<s 𝑅))
16	14, 15	anbi12d 644	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅)))
17	3, 4	jca 521	. . . . . 6 ⊢ (𝜑 → (𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅))
18	16, 5, 17	elrabd 3650	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
19		fnfvima 7236	. . . . 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 1494	. . . 4 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
21		intss1 4926	. . . 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 3968	. 2 ⊢ (𝜑 → ( bday ‘(𝐿 |s 𝑅)) ⊆ ( bday ‘𝐵))
24	2, 23	eqsstrd 3968	1 ⊢ (𝜑 → ( bday ‘𝐴) ⊆ ( bday ‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  {crab 3414   ⊆ wss 3902  ∅c0 4282  {csn 4587  ∩ cint 4910   class class class wbr 5107   “ cima 5662   Fn wfn 6532  ‘cfv 6537  (class class class)co 7417   No csur 27884   bday cbday 27886   <<s cslts 28030   |s ccuts 28032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1o 8459  df-2o 8460  df-no 27887  df-lts 27888  df-bday 27889  df-slts 28031  df-cuts 28033

This theorem is used by:  bdayfinbndlem1  28740