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Mirrors > Home > MPE Home > Th. List > 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 27775 | . . . 4 ⊢ 0s = (∅ |s ∅) | |
2 | snelpwi 5447 | . . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No ) | |
3 | nulsslt 27748 | . . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋}) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ No → ∅ <<s {𝑋}) |
5 | 4 | adantr 479 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∅ <<s {𝑋}) |
6 | nulssgt 27749 | . . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅) | |
7 | 2, 6 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} <<s ∅) |
8 | 7 | adantr 479 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → {𝑋} <<s ∅) |
9 | id 22 | . . . . . . . . 9 ⊢ (( bday ‘𝑋) = ∅ → ( bday ‘𝑋) = ∅) | |
10 | 0ss 4398 | . . . . . . . . 9 ⊢ ∅ ⊆ ( bday ‘𝑥) | |
11 | 9, 10 | eqsstrdi 4034 | . . . . . . . 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 480 | . . . . . 6 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))) |
14 | 13 | ralrimivw 3146 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))) |
15 | 0elpw 5358 | . . . . . . . 8 ⊢ ∅ ∈ 𝒫 No | |
16 | nulssgt 27749 | . . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
17 | 15, 16 | ax-mp 5 | . . . . . . 7 ⊢ ∅ <<s ∅ |
18 | eqscut2 27757 | . . . . . . 7 ⊢ ((∅ <<s ∅ ∧ 𝑋 ∈ No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))))) | |
19 | 17, 18 | mpan 688 | . . . . . 6 ⊢ (𝑋 ∈ No → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))))) |
20 | 19 | adantr 479 | . . . . 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 | eqtr2id 2780 | . . 3 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → 𝑋 = 0s ) |
23 | 22 | ex 411 | . 2 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ → 𝑋 = 0s )) |
24 | fveq2 6900 | . . 3 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ( bday ‘ 0s )) | |
25 | bday0s 27779 | . . 3 ⊢ ( bday ‘ 0s ) = ∅ | |
26 | 24, 25 | eqtrdi 2783 | . 2 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ∅) |
27 | 23, 26 | impbid1 224 | 1 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3057 ⊆ wss 3947 ∅c0 4324 𝒫 cpw 4604 {csn 4630 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 No csur 27591 bday cbday 27593 <<s csslt 27731 |s cscut 27733 0s c0s 27773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-ord 6375 df-on 6376 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1o 8491 df-2o 8492 df-no 27594 df-slt 27595 df-bday 27596 df-sslt 27732 df-scut 27734 df-0s 27775 |
This theorem is referenced by: bday1s 27782 cuteq1 27784 0elold 27853 |
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