Theorem cutbdaybnd2 28026

Description: An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.)

Assertion

Ref	Expression
cutbdaybnd2	⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))

Proof of Theorem cutbdaybnd2

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

Step	Hyp	Ref	Expression
1		etaslts2 28024	. 2 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
2		cutbday 28014	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
3	2	adantr 486	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
4		bdayfn 27978	. . . . . . 7 ⊢ bday Fn No
5		ssrab2 4037	. . . . . . 7 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
6		simprl 783	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → 𝑥 ∈ No )
7		simprr1 1240	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → 𝐴 <<s {𝑥})
8		simprr2 1241	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → {𝑥} <<s 𝐵)
9	7, 8	jca 521	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))
10		sneq 4604	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
11	10	breq2d 5126	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑥}))
12	10	breq1d 5124	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ({𝑦} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
13	11, 12	anbi12d 644	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
14	13	elrab 3653	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
15	6, 9, 14	sylanbrc 595	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → 𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
16		fnfvima 7238	. . . . . . 7 ⊢ (( 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	4, 5, 15, 16	mp3an12i 1494	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → ( bday ‘𝑥) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
18		intss1 4933	. . . . . 6 ⊢ (( 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	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑥))
20	3, 19	eqsstrd 3974	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑥))
21		simprr3 1242	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
22	20, 21	sstrd 3950	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
23	22	rexlimdvaa 3170	. 2 ⊢ (𝐴 <<s 𝐵 → (∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
24	1, 23	mpd 16	1 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  {crab 3419   ∪ cun 3906   ⊆ wss 3908  {csn 4594  ∪ cuni 4877  ∩ cint 4917   class class class wbr 5114   “ cima 5669  suc csuc 6369   Fn wfn 6538  ‘cfv 6543  (class class class)co 7423   No csur 27841   bday cbday 27843   <<s cslts 27987   |s ccuts 27989

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1o 8462  df-2o 8463  df-no 27844  df-lts 27845  df-bday 27846  df-slts 27988  df-cuts 27990

This theorem is used by:  cutbdaybnd2lim  28027  bday1  28044