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Theorem scutbdaybnd 27735
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 27733 . 2 ((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))
2 simpl1 1189 . . . . 5 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝐴 <<s 𝐵)
3 scutbday 27724 . . . . 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 27693 . . . . . 6 bday Fn No
6 ssrab2 4073 . . . . . 6 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
7 sneq 4634 . . . . . . . . 9 (𝑦 = 𝑥 → {𝑦} = {𝑥})
87breq2d 5154 . . . . . . . 8 (𝑦 = 𝑥 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑥}))
97breq1d 5152 . . . . . . . 8 (𝑦 = 𝑥 → ({𝑦} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
108, 9anbi12d 630 . . . . . . 7 (𝑦 = 𝑥 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
11 simprl 770 . . . . . . 7 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝑥 No )
12 simprr1 1219 . . . . . . . 8 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝐴 <<s {𝑥})
13 simprr2 1220 . . . . . . . 8 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → {𝑥} <<s 𝐵)
1412, 13jca 511 . . . . . . 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 3682 . . . . . 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 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)}))
175, 6, 15, 16mp3an12i 1462 . . . . 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 4961 . . . . 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 4016 . . 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 1221 . . 3 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → ( bday 𝑥) ⊆ 𝑂)
2220, 21sstrd 3988 . 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 3156 1 ((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  {crab 3427  cun 3942  wss 3944  {csn 4624   cint 4944   class class class wbr 5142  cima 5675  Oncon0 6363   Fn wfn 6537  cfv 6542  (class class class)co 7414   No csur 27560   bday cbday 27562   <<s csslt 27700   |s cscut 27702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1o 8480  df-2o 8481  df-no 27563  df-slt 27564  df-bday 27565  df-sslt 27701  df-scut 27703
This theorem is referenced by:  madebdayim  27801  negsbdaylem  27955  n0sbday  28204
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