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Theorem scutbdaybnd2 27551
Description: An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.)
Assertion
Ref Expression
scutbdaybnd2 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))

Proof of Theorem scutbdaybnd2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 etasslt2 27549 . 2 (𝐴 <<s 𝐵 → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))
2 scutbday 27539 . . . . . 6 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
32adantr 480 . . . . 5 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
4 bdayfn 27508 . . . . . . 7 bday Fn No
5 ssrab2 4078 . . . . . . 7 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
6 simprl 768 . . . . . . . 8 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → 𝑥 No )
7 simprr1 1220 . . . . . . . . 9 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → 𝐴 <<s {𝑥})
8 simprr2 1221 . . . . . . . . 9 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → {𝑥} <<s 𝐵)
97, 8jca 511 . . . . . . . 8 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))
10 sneq 4639 . . . . . . . . . . 11 (𝑦 = 𝑥 → {𝑦} = {𝑥})
1110breq2d 5161 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑥}))
1210breq1d 5159 . . . . . . . . . 10 (𝑦 = 𝑥 → ({𝑦} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
1311, 12anbi12d 630 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
1413elrab 3684 . . . . . . . 8 (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
156, 9, 14sylanbrc 582 . . . . . . 7 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → 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)}))
174, 5, 15, 16mp3an12i 1464 . . . . . 6 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → ( bday 𝑥) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
18 intss1 4968 . . . . . 6 (( 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 . . . . 5 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday 𝑥))
203, 19eqsstrd 4021 . . . 4 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑥))
21 simprr3 1222 . . . 4 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵)))
2220, 21sstrd 3993 . . 3 ((𝐴 <<s 𝐵 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
2322rexlimdvaa 3155 . 2 (𝐴 <<s 𝐵 → (∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵))))
241, 23mpd 15 1 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069  {crab 3431  cun 3947  wss 3949  {csn 4629   cuni 4909   cint 4951   class class class wbr 5149  cima 5680  suc csuc 6367   Fn wfn 6539  cfv 6544  (class class class)co 7412   No csur 27376   bday cbday 27378   <<s csslt 27515   |s cscut 27517
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1o 8469  df-2o 8470  df-no 27379  df-slt 27380  df-bday 27381  df-sslt 27516  df-scut 27518
This theorem is referenced by:  scutbdaybnd2lim  27552  bday1s  27566
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