Theorem cutbdaybnd 28214

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 28212	. 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 28203	. . . . 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 28167	. . . . . 6 ⊢ bday Fn No
6		ssrab2 4008	. . . . . 6 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
7		sneq 4594	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
8	7	breq2d 5118	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑥}))
9	7	breq1d 5116	. . . . . . . 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 3650	. . . . . 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 7265	. . . . . 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 1495	. . . . 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 4926	. . . . 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 3947	. . 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 3923	. 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 3173	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 1570   ∈ wcel 2147  {crab 3416   ⊆ wss 3900   ∪ cun 4152  {csn 4588  ∩ cint 4910   class class class wbr 5106   “ cima 5659  Oncon0 6383   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  Nocsur 28030  bdaycbday 28032   <<s cslts 28176   |s ccuts 28178

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179

This theorem is used by:  madebdayim  28307  bdayiun  28334  addbday  28437  negbdaylem  28475  oncutlt  28683  oniso  28690  bdayons  28695  n0bday  28771  bdayn0p1  28788  bdaypw2n0bndlem  28882