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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 27251 | . . . 4 ⊢ 0s = (∅ |s ∅) | |
2 | 1 | fveq2i 6881 | . . 3 ⊢ ( bday ‘ 0s ) = ( bday ‘(∅ |s ∅)) |
3 | 0elpw 5347 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
4 | nulssgt 27225 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
5 | scutbday 27231 | . . . 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 2759 | . 2 ⊢ ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) |
8 | snelpwi 5436 | . . . . . . . 8 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No ) | |
9 | nulsslt 27224 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝒫 No → ∅ <<s {𝑥}) | |
10 | nulssgt 27225 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝒫 No → {𝑥} <<s ∅) | |
11 | 9, 10 | jca 512 | . . . . . . . 8 ⊢ ({𝑥} ∈ 𝒫 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)) |
12 | 8, 11 | syl 17 | . . . . . . 7 ⊢ (𝑥 ∈ No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)) |
13 | 12 | rabeqc 3443 | . . . . . 6 ⊢ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = No |
14 | bdaydm 27202 | . . . . . 6 ⊢ dom bday = No | |
15 | 13, 14 | eqtr4i 2762 | . . . . 5 ⊢ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = dom bday |
16 | 15 | imaeq2i 6047 | . . . 4 ⊢ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ( bday “ dom bday ) |
17 | imadmrn 6059 | . . . 4 ⊢ ( bday “ dom bday ) = ran bday | |
18 | bdayrn 27203 | . . . 4 ⊢ ran bday = On | |
19 | 16, 17, 18 | 3eqtri 2763 | . . 3 ⊢ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On |
20 | 19 | inteqi 4947 | . 2 ⊢ ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ∩ On |
21 | inton 6411 | . 2 ⊢ ∩ On = ∅ | |
22 | 7, 20, 21 | 3eqtri 2763 | 1 ⊢ ( bday ‘ 0s ) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3431 ∅c0 4318 𝒫 cpw 4596 {csn 4622 ∩ cint 4943 class class class wbr 5141 dom cdm 5669 ran crn 5670 “ cima 5672 Oncon0 6353 ‘cfv 6532 (class class class)co 7393 No csur 27070 bday cbday 27072 <<s csslt 27208 |s cscut 27210 0s c0s 27249 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 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 6356 df-on 6357 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1o 8448 df-2o 8449 df-no 27073 df-slt 27074 df-bday 27075 df-sslt 27209 df-scut 27211 df-0s 27251 |
This theorem is referenced by: bday0b 27257 bday1s 27258 cuteq0 27259 left0s 27310 right0s 27311 addsproplem2 27370 negsproplem2 27419 negsproplem6 27423 mulsproplem2 27486 mulsproplem3 27487 mulsproplem4 27488 mulsproplem5 27489 mulsproplem6 27490 mulsproplem7 27491 mulsproplem8 27492 mulsproplem12 27496 mulsproplem13 27497 mulsproplem14 27498 |
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