Theorem scutbdaybnd 27749

Description: An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Aug-2024.)

Assertion

Ref	Expression
scutbdaybnd	⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)

Proof of Theorem scutbdaybnd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etasslt 27747	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))
2		simpl1 1192	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝐴 <<s 𝐵)
3		scutbday 27738	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
4	2, 3	syl 17	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
5		bdayfn 27705	. . . . . 6 ⊢ bday Fn No
6		ssrab2 4028	. . . . . 6 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
7		sneq 4584	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
8	7	breq2d 5101	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑥}))
9	7	breq1d 5099	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ({𝑦} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
10	8, 9	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
11		simprl 770	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝑥 ∈ No )
12		simprr1 1222	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝐴 <<s {𝑥})
13		simprr2 1223	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → {𝑥} <<s 𝐵)
14	12, 13	jca 511	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))
15	10, 11, 14	elrabd 3647	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
16		fnfvima 7162	. . . . . 6 ⊢ (( bday Fn No ∧ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No ∧ 𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday ‘𝑥) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
17	5, 6, 15, 16	mp3an12i 1467	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ( bday ‘𝑥) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
18		intss1 4911	. . . . 5 ⊢ (( bday ‘𝑥) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑥))
19	17, 18	syl 17	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑥))
20	4, 19	eqsstrd 3967	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑥))
21		simprr3 1224	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ( bday ‘𝑥) ⊆ 𝑂)
22	20, 21	sstrd 3943	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)
23	1, 22	rexlimddv 3137	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  {crab 3393   ∪ cun 3898   ⊆ wss 3900  {csn 4574  ∩ cint 4895   class class class wbr 5089   “ cima 5617  Oncon0 6302   Fn wfn 6472  ‘cfv 6477  (class class class)co 7341   No csur 27571   bday cbday 27573   <<s csslt 27713   |s cscut 27715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6305  df-on 6306  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1o 8380  df-2o 8381  df-no 27574  df-slt 27575  df-bday 27576  df-sslt 27714  df-scut 27716

This theorem is referenced by:  madebdayim  27826  bdayiun  27853  addsbday  27953  negsbdaylem  27991  onscutlt  28194  onsiso  28198  bdayon  28202  n0sbday  28273  bdayn0p1  28287  zs12bday  28387