Theorem cutbdaybnd 28186

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

Assertion

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

Proof of Theorem cutbdaybnd

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

Step	Hyp	Ref	Expression
1		etaslts 28184	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ (bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ (bday‘𝑥) ⊆ 𝑂))
2		simpl1 1210	. . . . 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		cutbday 28175	. . . . 5 ⊢ (𝐴 <<s 𝐵 → (bday‘(𝐴 |s 𝐵)) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
4	2, 3	syl 18	. . . 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 28139	. . . . . 6 ⊢ bday Fn No
6		ssrab2 4005	. . . . . 6 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
7		sneq 4590	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
8	7	breq2d 5114	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑥}))
9	7	breq1d 5112	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ({𝑦} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
10	8, 9	anbi12d 644	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
11		simprl 783	. . . . . . 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 1240	. . . . . . . 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 1241	. . . . . . . 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 521	. . . . . . 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 3646	. . . . . 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 7239	. . . . . 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 1494	. . . . 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 4922	. . . . 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 18	. . . 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 3942	. . 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 1242	. . 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 3918	. 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 3169	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ (bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → (bday‘(𝐴 |s 𝐵)) ⊆ 𝑂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3895   ∪ cun 4149  {csn 4584  ∩ cint 4906   class class class wbr 5102   “ cima 5653  Oncon0 6362   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Nocsur 28002  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151

This theorem is used by:  madebdayim  28279  bdayiun  28306  addbday  28409  negbdaylem  28447  oncutlt  28655  oniso  28662  bdayons  28667  n0bday  28743  bdayn0p1  28760  bdaypw2n0bndlem  28854