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Theorem scutbdaybnd 27324
Description: An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Aug-2024.)
Assertion
Ref Expression
scutbdaybnd ((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)

Proof of Theorem scutbdaybnd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 etasslt 27322 . 2 ((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))
2 simpl1 1191 . . . . 5 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝐴 <<s 𝐵)
3 scutbday 27313 . . . . 5 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
42, 3syl 17 . . . 4 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 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 27282 . . . . . 6 bday Fn No
6 ssrab2 4077 . . . . . 6 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
7 sneq 4638 . . . . . . . . 9 (𝑦 = 𝑥 → {𝑦} = {𝑥})
87breq2d 5160 . . . . . . . 8 (𝑦 = 𝑥 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑥}))
97breq1d 5158 . . . . . . . 8 (𝑦 = 𝑥 → ({𝑦} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
108, 9anbi12d 631 . . . . . . 7 (𝑦 = 𝑥 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
11 simprl 769 . . . . . . 7 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝑥 No )
12 simprr1 1221 . . . . . . . 8 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝐴 <<s {𝑥})
13 simprr2 1222 . . . . . . . 8 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → {𝑥} <<s 𝐵)
1412, 13jca 512 . . . . . . 7 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))
1510, 11, 14elrabd 3685 . . . . . 6 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
16 fnfvima 7237 . . . . . 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)}))
175, 6, 15, 16mp3an12i 1465 . . . . 5 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 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 4967 . . . . 5 (( bday 𝑥) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday 𝑥))
1917, 18syl 17 . . . 4 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday 𝑥))
204, 19eqsstrd 4020 . . 3 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑥))
21 simprr3 1223 . . 3 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → ( bday 𝑥) ⊆ 𝑂)
2220, 21sstrd 3992 . 2 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)
231, 22rexlimddv 3161 1 ((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {crab 3432  cun 3946  wss 3948  {csn 4628   cint 4950   class class class wbr 5148  cima 5679  Oncon0 6364   Fn wfn 6538  cfv 6543  (class class class)co 7411   No csur 27150   bday cbday 27152   <<s csslt 27289   |s cscut 27291
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1o 8468  df-2o 8469  df-no 27153  df-slt 27154  df-bday 27155  df-sslt 27290  df-scut 27292
This theorem is referenced by:  madebdayim  27390  negsbdaylem  27540
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