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Theorem scutbday 27692
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 27688 . . 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 2732 . 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 27689 . . . 4 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4 sneq 4633 . . . . . . . 8 (𝑥 = (𝐴 |s 𝐵) → {𝑥} = {(𝐴 |s 𝐵)})
54breq2d 5153 . . . . . . 7 (𝑥 = (𝐴 |s 𝐵) → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {(𝐴 |s 𝐵)}))
64breq1d 5151 . . . . . . 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 3678 . . . . 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 278 . . . 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 27687 . . 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 6894 . . . 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 7387 . . 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 583 . 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 257 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 395  w3a 1084   = wceq 1533  wcel 2098  ∃!wreu 3368  {crab 3426  {csn 4623   cint 4943   class class class wbr 5141  cima 5672  cfv 6537  crio 7360  (class class class)co 7405   No csur 27528   bday cbday 27530   <<s csslt 27668   |s cscut 27670
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1o 8467  df-2o 8468  df-no 27531  df-slt 27532  df-bday 27533  df-sslt 27669  df-scut 27671
This theorem is referenced by:  scutun12  27698  scutbdaybnd  27703  scutbdaybnd2  27704  scutbdaylt  27706  bday0s  27716  bday1s  27719  cofcut1  27795  cofcutr  27799
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