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Theorem scutbdaybnd 27176
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 27174 . 2 ((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))
2 simpl1 1192 . . . . 5 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝐴 <<s 𝐵)
3 scutbday 27165 . . . . 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 27135 . . . . . 6 bday Fn No
6 ssrab2 4038 . . . . . 6 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
7 sneq 4597 . . . . . . . . 9 (𝑦 = 𝑥 → {𝑦} = {𝑥})
87breq2d 5118 . . . . . . . 8 (𝑦 = 𝑥 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑥}))
97breq1d 5116 . . . . . . . 8 (𝑦 = 𝑥 → ({𝑦} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
108, 9anbi12d 632 . . . . . . 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 1222 . . . . . . . 8 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → 𝐴 <<s {𝑥})
13 simprr2 1223 . . . . . . . 8 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → {𝑥} <<s 𝐵)
1412, 13jca 513 . . . . . . 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 3648 . . . . . 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 7184 . . . . . 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 1466 . . . . 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 4925 . . . . 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 3983 . . 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 1224 . . 3 (((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))) → ( bday 𝑥) ⊆ 𝑂)
2220, 21sstrd 3955 . 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 3155 1 ((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  {crab 3406  cun 3909  wss 3911  {csn 4587   cint 4908   class class class wbr 5106  cima 5637  Oncon0 6318   Fn wfn 6492  cfv 6497  (class class class)co 7358   No csur 27004   bday cbday 27006   <<s csslt 27142   |s cscut 27144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009  df-sslt 27143  df-scut 27145
This theorem is referenced by:  madebdayim  27239
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