Theorem bday0s 33616

Description: Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.)

Assertion

Ref	Expression
bday0s	⊢ ( bday ‘ 0s ) = ∅

Proof of Theorem bday0s

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-0s 33612	. . . 4 ⊢ 0s = (∅ |s ∅)
2	1	fveq2i 6666	. . 3 ⊢ ( bday ‘ 0s ) = ( bday ‘(∅ |s ∅))
3		0elpw 5228	. . . 4 ⊢ ∅ ∈ 𝒫 No
4		nulssgt 33587	. . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
5		scutbday 33593	. . . 4 ⊢ (∅ <<s ∅ → ( bday ‘(∅ |s ∅)) = ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}))
6	3, 4, 5	mp2b 10	. . 3 ⊢ ( bday ‘(∅ |s ∅)) = ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)})
7	2, 6	eqtri 2781	. 2 ⊢ ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)})
8		snelpwi 5309	. . . . . . . 8 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No )
9		nulsslt 33586	. . . . . . . . 9 ⊢ ({𝑥} ∈ 𝒫 No → ∅ <<s {𝑥})
10		nulssgt 33587	. . . . . . . . 9 ⊢ ({𝑥} ∈ 𝒫 No → {𝑥} <<s ∅)
11	9, 10	jca 515	. . . . . . . 8 ⊢ ({𝑥} ∈ 𝒫 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅))
12	8, 11	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅))
13	12	rabeqc 3602	. . . . . 6 ⊢ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = No
14		bdaydm 33566	. . . . . 6 ⊢ dom bday = No
15	13, 14	eqtr4i 2784	. . . . 5 ⊢ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = dom bday
16	15	imaeq2i 5904	. . . 4 ⊢ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ( bday “ dom bday )
17		imadmrn 5916	. . . 4 ⊢ ( bday “ dom bday ) = ran bday
18		bdayrn 33567	. . . 4 ⊢ ran bday = On
19	16, 17, 18	3eqtri 2785	. . 3 ⊢ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On
20	19	inteqi 4845	. 2 ⊢ ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ∩ On
21		inton 6231	. 2 ⊢ ∩ On = ∅
22	7, 20, 21	3eqtri 2785	1 ⊢ ( bday ‘ 0s ) = ∅

Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074  ∅c0 4227  𝒫 cpw 4497  {csn 4525  ∩ cint 4841   class class class wbr 5036  dom cdm 5528  ran crn 5529   “ cima 5531  Oncon0 6174  ‘cfv 6340  (class class class)co 7156   No csur 33440   bday cbday 33442   <<s csslt 33572   |s cscut 33574   0s c0s 33610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1o 8118  df-2o 8119  df-no 33443  df-slt 33444  df-bday 33445  df-sslt 33573  df-scut 33575  df-0s 33612

This theorem is referenced by:  bday0b  33618  bday1s  33619  left0s  33666  right0s  33667