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Mirrors > Home > MPE Home > Th. List > bday0s | Structured version Visualization version GIF version |
Description: Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.) |
Ref | Expression |
---|---|
bday0s | ⊢ ( bday ‘ 0s ) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0s 27185 | . . . 4 ⊢ 0s = (∅ |s ∅) | |
2 | 1 | fveq2i 6846 | . . 3 ⊢ ( bday ‘ 0s ) = ( bday ‘(∅ |s ∅)) |
3 | 0elpw 5312 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
4 | nulssgt 27159 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
5 | scutbday 27165 | . . . 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 2761 | . 2 ⊢ ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) |
8 | snelpwi 5401 | . . . . . . . 8 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No ) | |
9 | nulsslt 27158 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝒫 No → ∅ <<s {𝑥}) | |
10 | nulssgt 27159 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝒫 No → {𝑥} <<s ∅) | |
11 | 9, 10 | jca 513 | . . . . . . . 8 ⊢ ({𝑥} ∈ 𝒫 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)) |
12 | 8, 11 | syl 17 | . . . . . . 7 ⊢ (𝑥 ∈ No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)) |
13 | 12 | rabeqc 3418 | . . . . . 6 ⊢ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = No |
14 | bdaydm 27136 | . . . . . 6 ⊢ dom bday = No | |
15 | 13, 14 | eqtr4i 2764 | . . . . 5 ⊢ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = dom bday |
16 | 15 | imaeq2i 6012 | . . . 4 ⊢ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ( bday “ dom bday ) |
17 | imadmrn 6024 | . . . 4 ⊢ ( bday “ dom bday ) = ran bday | |
18 | bdayrn 27137 | . . . 4 ⊢ ran bday = On | |
19 | 16, 17, 18 | 3eqtri 2765 | . . 3 ⊢ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On |
20 | 19 | inteqi 4912 | . 2 ⊢ ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ∩ On |
21 | inton 6376 | . 2 ⊢ ∩ On = ∅ | |
22 | 7, 20, 21 | 3eqtri 2765 | 1 ⊢ ( bday ‘ 0s ) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 ∅c0 4283 𝒫 cpw 4561 {csn 4587 ∩ cint 4908 class class class wbr 5106 dom cdm 5634 ran crn 5635 “ cima 5637 Oncon0 6318 ‘cfv 6497 (class class class)co 7358 No csur 27004 bday cbday 27006 <<s csslt 27142 |s cscut 27144 0s c0s 27183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1o 8413 df-2o 8414 df-no 27007 df-slt 27008 df-bday 27009 df-sslt 27143 df-scut 27145 df-0s 27185 |
This theorem is referenced by: bday0b 27191 bday1s 27192 cuteq0 27193 left0s 27244 right0s 27245 addsproplem2 27304 negsproplem2 27349 negsproplem6 27353 mulsproplem2 27402 mulsproplem3 27403 mulsproplem4 27404 mulsproplem5 27405 |
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