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Theorem scutbday 32219
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 32217 . . 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 2766 . 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 32218 . . . 4 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4 sneq 4331 . . . . . . . 8 (𝑥 = (𝐴 |s 𝐵) → {𝑥} = {(𝐴 |s 𝐵)})
54breq2d 4816 . . . . . . 7 (𝑥 = (𝐴 |s 𝐵) → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {(𝐴 |s 𝐵)}))
64breq1d 4814 . . . . . . 7 (𝑥 = (𝐴 |s 𝐵) → ({𝑥} <<s 𝐵 ↔ {(𝐴 |s 𝐵)} <<s 𝐵))
75, 6anbi12d 749 . . . . . 6 (𝑥 = (𝐴 |s 𝐵) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
87elrab 3504 . . . . 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 1081 . . . . 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 267 . . . 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 224 . . 3 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})
12 conway 32216 . . 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 fveq2 6352 . . . . 5 (𝑦 = (𝐴 |s 𝐵) → ( bday 𝑦) = ( bday ‘(𝐴 |s 𝐵)))
1413eqeq1d 2762 . . . 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)})))
1514riota2 6796 . . 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)))
1611, 12, 15syl2anc 696 . 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)))
172, 16mpbird 247 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  ∃!wreu 3052  {crab 3054  {csn 4321   cint 4627   class class class wbr 4804  cima 5269  cfv 6049  crio 6773  (class class class)co 6813   No csur 32099   bday cbday 32101   <<s csslt 32202   |s cscut 32204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1o 7729  df-2o 7730  df-no 32102  df-slt 32103  df-bday 32104  df-sslt 32203  df-scut 32205
This theorem is referenced by:  scutun12  32223  scutbdaybnd  32227  scutbdaylt  32228
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