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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 27326 | . . . 4 ⊢ 0s = (∅ |s ∅) | |
2 | 1 | fveq2i 6895 | . . 3 ⊢ ( bday ‘ 0s ) = ( bday ‘(∅ |s ∅)) |
3 | 0elpw 5355 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
4 | nulssgt 27300 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
5 | scutbday 27306 | . . . 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 5444 | . . . . . . . 8 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No ) | |
9 | nulsslt 27299 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝒫 No → ∅ <<s {𝑥}) | |
10 | nulssgt 27300 | . . . . . . . . 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 3445 | . . . . . 6 ⊢ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = No |
14 | bdaydm 27276 | . . . . . 6 ⊢ dom bday = No | |
15 | 13, 14 | eqtr4i 2764 | . . . . 5 ⊢ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = dom bday |
16 | 15 | imaeq2i 6058 | . . . 4 ⊢ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ( bday “ dom bday ) |
17 | imadmrn 6070 | . . . 4 ⊢ ( bday “ dom bday ) = ran bday | |
18 | bdayrn 27277 | . . . 4 ⊢ ran bday = On | |
19 | 16, 17, 18 | 3eqtri 2765 | . . 3 ⊢ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On |
20 | 19 | inteqi 4955 | . 2 ⊢ ∩ ( bday “ {𝑥 ∈ No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ∩ On |
21 | inton 6423 | . 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 3433 ∅c0 4323 𝒫 cpw 4603 {csn 4629 ∩ cint 4951 class class class wbr 5149 dom cdm 5677 ran crn 5678 “ cima 5680 Oncon0 6365 ‘cfv 6544 (class class class)co 7409 No csur 27143 bday cbday 27145 <<s csslt 27282 |s cscut 27284 0s c0s 27324 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1o 8466 df-2o 8467 df-no 27146 df-slt 27147 df-bday 27148 df-sslt 27283 df-scut 27285 df-0s 27326 |
This theorem is referenced by: bday0b 27332 bday1s 27333 cuteq0 27334 left0s 27388 right0s 27389 0elold 27404 addsproplem2 27456 negsproplem2 27506 negsproplem6 27510 mulsproplem2 27576 mulsproplem3 27577 mulsproplem4 27578 mulsproplem5 27579 mulsproplem6 27580 mulsproplem7 27581 mulsproplem8 27582 mulsproplem12 27586 mulsproplem13 27587 mulsproplem14 27588 |
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