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Theorem scutbday 27731

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.

Step	Hyp	Ref	Expression
1		scutval 27727	. . 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)})))
2	1	eqcomd 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 27728	. . . 4 ⊢ (𝐴 <<s 𝐵 → ((𝐴 \|s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 \|s 𝐵)} ∧ {(𝐴 \|s 𝐵)} <<s 𝐵))
4		sneq 4635	. . . . . . . 8 ⊢ (𝑥 = (𝐴 \|s 𝐵) → {𝑥} = {(𝐴 \|s 𝐵)})
5	4	breq2d 5155	. . . . . . 7 ⊢ (𝑥 = (𝐴 \|s 𝐵) → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {(𝐴 \|s 𝐵)}))
6	4	breq1d 5153	. . . . . . 7 ⊢ (𝑥 = (𝐴 \|s 𝐵) → ({𝑥} <<s 𝐵 ↔ {(𝐴 \|s 𝐵)} <<s 𝐵))
7	5, 6	anbi12d 631	. . . . . 6 ⊢ (𝑥 = (𝐴 \|s 𝐵) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {(𝐴 \|s 𝐵)} ∧ {(𝐴 \|s 𝐵)} <<s 𝐵)))
8	7	elrab 3681	. . . . 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 1093	. . . . 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)))
10	8, 9	bitr4i 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))
11	3, 10	sylibr 233	. . 3 ⊢ (𝐴 <<s 𝐵 → (𝐴 \|s 𝐵) ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})
12		conway 27726	. . 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 6901	. . . 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)})))
14	13	riota2 7397	. . 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)))
15	11, 12, 14	syl2anc 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)))
16	2, 15	mpbird 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 1085 = wceq 1534 ∈ wcel 2099 ∃!wreu 3370 {crab 3428 {csn 4625 ∩ cint 4945 class class class wbr 5143 “ cima 5676 ‘cfv 6543 ℩crio 7370 (class class class)co 7415 No csur 27567 bday cbday 27569 <<s csslt 27707 |s cscut 27709

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 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-un 7735

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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1o 8481 df-2o 8482 df-no 27570 df-slt 27571 df-bday 27572 df-sslt 27708 df-scut 27710

This theorem is referenced by: scutun12 27737 scutbdaybnd 27742 scutbdaybnd2 27743 scutbdaylt 27745 bday0s 27755 bday1s 27758 cofcut1 27834 cofcutr 27838

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