Theorem sltsbday 27934

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 6838	. 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 4714	. . . . 5 ⊢ (𝜑 → {𝐵} ≠ ∅)
7		sltstr 27804	. . . . 5 ⊢ ((𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅 ∧ {𝐵} ≠ ∅) → 𝐿 <<s 𝑅)
8	3, 4, 6, 7	syl3anc 1379	. . . 4 ⊢ (𝜑 → 𝐿 <<s 𝑅)
9		cutbday 27801	. . . 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 27766	. . . . 5 ⊢ bday Fn No
12		ssrab2 4018	. . . . 5 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
13		sneq 4572	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
14	13	breq2d 5091	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝐵}))
15	13	breq1d 5089	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ({𝑥} <<s 𝑅 ↔ {𝐵} <<s 𝑅))
16	14, 15	anbi12d 638	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅)))
17	3, 4	jca 516	. . . . . 6 ⊢ (𝜑 → (𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅))
18	16, 5, 17	elrabd 3638	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
19		fnfvima 7184	. . . . 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 1473	. . . 4 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
21		intss1 4900	. . . 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 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  {crab 3392   ⊆ wss 3890  ∅c0 4268  {csn 4562  ∩ cint 4884   class class class wbr 5079   “ cima 5628   Fn wfn 6487  ‘cfv 6492  (class class class)co 7363   No csur 27628   bday cbday 27630   <<s cslts 27774   |s ccuts 27776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1o 8402  df-2o 8403  df-no 27631  df-lts 27632  df-bday 27633  df-slts 27775  df-cuts 27777

This theorem is referenced by:  bdayfinbndlem1  28484