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Mirrors > Home > MPE Home > Th. List > Mathboxes > bday0b | Structured version Visualization version GIF version |
Description: The only surreal with birthday ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.) |
Ref | Expression |
---|---|
bday0b | ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0s 33604 | . . . 4 ⊢ 0s = (∅ |s ∅) | |
2 | snelpwi 5308 | . . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No ) | |
3 | nulsslt 33578 | . . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋}) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ No → ∅ <<s {𝑋}) |
5 | 4 | adantr 484 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∅ <<s {𝑋}) |
6 | nulssgt 33579 | . . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅) | |
7 | 2, 6 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} <<s ∅) |
8 | 7 | adantr 484 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → {𝑋} <<s ∅) |
9 | id 22 | . . . . . . . . 9 ⊢ (( bday ‘𝑋) = ∅ → ( bday ‘𝑋) = ∅) | |
10 | 0ss 4295 | . . . . . . . . 9 ⊢ ∅ ⊆ ( bday ‘𝑥) | |
11 | 9, 10 | eqsstrdi 3948 | . . . . . . . 8 ⊢ (( bday ‘𝑋) = ∅ → ( bday ‘𝑋) ⊆ ( bday ‘𝑥)) |
12 | 11 | a1d 25 | . . . . . . 7 ⊢ (( bday ‘𝑋) = ∅ → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))) |
13 | 12 | adantl 485 | . . . . . 6 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))) |
14 | 13 | ralrimivw 3114 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))) |
15 | 0elpw 5227 | . . . . . . . 8 ⊢ ∅ ∈ 𝒫 No | |
16 | nulssgt 33579 | . . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
17 | 15, 16 | ax-mp 5 | . . . . . . 7 ⊢ ∅ <<s ∅ |
18 | eqscut2 33587 | . . . . . . 7 ⊢ ((∅ <<s ∅ ∧ 𝑋 ∈ No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))))) | |
19 | 17, 18 | mpan 689 | . . . . . 6 ⊢ (𝑋 ∈ No → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))))) |
20 | 19 | adantr 484 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))))) |
21 | 5, 8, 14, 20 | mpbir3and 1339 | . . . 4 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → (∅ |s ∅) = 𝑋) |
22 | 1, 21 | syl5req 2806 | . . 3 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → 𝑋 = 0s ) |
23 | 22 | ex 416 | . 2 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ → 𝑋 = 0s )) |
24 | fveq2 6662 | . . 3 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ( bday ‘ 0s )) | |
25 | bday0s 33608 | . . 3 ⊢ ( bday ‘ 0s ) = ∅ | |
26 | 24, 25 | eqtrdi 2809 | . 2 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ∅) |
27 | 23, 26 | impbid1 228 | 1 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3070 ⊆ wss 3860 ∅c0 4227 𝒫 cpw 4497 {csn 4525 class class class wbr 5035 ‘cfv 6339 (class class class)co 7155 No csur 33432 bday cbday 33434 <<s csslt 33564 |s cscut 33566 0s c0s 33602 |
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 5159 ax-sep 5172 ax-nul 5179 ax-pr 5301 ax-un 7464 |
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 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-ord 6176 df-on 6177 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1o 8117 df-2o 8118 df-no 33435 df-slt 33436 df-bday 33437 df-sslt 33565 df-scut 33567 df-0s 33604 |
This theorem is referenced by: bday1s 33611 |
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