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Theorem scutbday 27765
Description: The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
scutbday (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem scutbday
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 scutval 27761 . . 3 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})))
21eqcomd 2734 . 2 (𝐴 <<s 𝐵 → (𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) = (𝐴 |s 𝐵))
3 scutcut 27762 . . . 4 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4 sneq 4642 . . . . . . . 8 (𝑥 = (𝐴 |s 𝐵) → {𝑥} = {(𝐴 |s 𝐵)})
54breq2d 5164 . . . . . . 7 (𝑥 = (𝐴 |s 𝐵) → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {(𝐴 |s 𝐵)}))
64breq1d 5162 . . . . . . 7 (𝑥 = (𝐴 |s 𝐵) → ({𝑥} <<s 𝐵 ↔ {(𝐴 |s 𝐵)} <<s 𝐵))
75, 6anbi12d 630 . . . . . 6 (𝑥 = (𝐴 |s 𝐵) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
87elrab 3684 . . . . 5 ((𝐴 |s 𝐵) ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ↔ ((𝐴 |s 𝐵) ∈ No ∧ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
9 3anass 1092 . . . . 5 (((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵) ↔ ((𝐴 |s 𝐵) ∈ No ∧ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
108, 9bitr4i 277 . . . 4 ((𝐴 |s 𝐵) ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ↔ ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
113, 10sylibr 233 . . 3 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})
12 conway 27760 . . 3 (𝐴 <<s 𝐵 → ∃!𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
13 fveqeq2 6911 . . . 4 (𝑦 = (𝐴 |s 𝐵) → (( bday 𝑦) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})))
1413riota2 7408 . . 3 (((𝐴 |s 𝐵) ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ∧ ∃!𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) → (( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ (𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) = (𝐴 |s 𝐵)))
1511, 12, 14syl2anc 582 . 2 (𝐴 <<s 𝐵 → (( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ (𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) = (𝐴 |s 𝐵)))
162, 15mpbird 256 1 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  ∃!wreu 3372  {crab 3430  {csn 4632   cint 4953   class class class wbr 5152  cima 5685  cfv 6553  crio 7381  (class class class)co 7426   No csur 27601   bday cbday 27603   <<s csslt 27741   |s cscut 27743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  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 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1o 8495  df-2o 8496  df-no 27604  df-slt 27605  df-bday 27606  df-sslt 27742  df-scut 27744
This theorem is referenced by:  scutun12  27771  scutbdaybnd  27776  scutbdaybnd2  27777  scutbdaylt  27779  bday0s  27789  bday1s  27792  cofcut1  27868  cofcutr  27872
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