Theorem ssltbday 27897

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

Hypotheses

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

Assertion

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

Proof of Theorem ssltbday

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssltbday.1	. . 3 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
2	1	fveq2d 6837	. 2 ⊢ (𝜑 → ( bday ‘𝐴) = ( bday ‘(𝐿 |s 𝑅)))
3		ssltbday.3	. . . . 5 ⊢ (𝜑 → 𝐿 <<s {𝐵})
4		ssltbday.4	. . . . 5 ⊢ (𝜑 → {𝐵} <<s 𝑅)
5		ssltbday.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
6	5	snn0d 4731	. . . . 5 ⊢ (𝜑 → {𝐵} ≠ ∅)
7		sslttr 27783	. . . . 5 ⊢ ((𝐿 <<s {𝐵} ∧ {𝐵} <<s 𝑅 ∧ {𝐵} ≠ ∅) → 𝐿 <<s 𝑅)
8	3, 4, 6, 7	syl3anc 1374	. . . 4 ⊢ (𝜑 → 𝐿 <<s 𝑅)
9		scutbday 27780	. . . 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 27747	. . . . 5 ⊢ bday Fn No
12		ssrab2 4031	. . . . 5 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
13		sneq 4589	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
14	13	breq2d 5109	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝐵}))
15	13	breq1d 5107	. . . . . . 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 3647	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
19		fnfvima 7179	. . . . 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 4917	. . . 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 3967	. 2 ⊢ (𝜑 → ( bday ‘(𝐿 |s 𝑅)) ⊆ ( bday ‘𝐵))
24	2, 23	eqsstrd 3967	1 ⊢ (𝜑 → ( bday ‘𝐴) ⊆ ( bday ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  {crab 3398   ⊆ wss 3900  ∅c0 4284  {csn 4579  ∩ cint 4901   class class class wbr 5097   “ cima 5626   Fn wfn 6486  ‘cfv 6491  (class class class)co 7358   No csur 27609   bday cbday 27611   <<s csslt 27755   |s cscut 27757

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6319  df-on 6320  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8397  df-2o 8398  df-no 27612  df-slt 27613  df-bday 27614  df-sslt 27756  df-scut 27758

This theorem is referenced by:  bdayfinbndlem1  28444