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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 27708 | . . . 4 ⊢ 0s = (∅ |s ∅) | |
2 | snelpwi 5436 | . . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No ) | |
3 | nulsslt 27681 | . . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋}) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ No → ∅ <<s {𝑋}) |
5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∅ <<s {𝑋}) |
6 | nulssgt 27682 | . . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅) | |
7 | 2, 6 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} <<s ∅) |
8 | 7 | adantr 480 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → {𝑋} <<s ∅) |
9 | id 22 | . . . . . . . . 9 ⊢ (( bday ‘𝑋) = ∅ → ( bday ‘𝑋) = ∅) | |
10 | 0ss 4391 | . . . . . . . . 9 ⊢ ∅ ⊆ ( bday ‘𝑥) | |
11 | 9, 10 | eqsstrdi 4031 | . . . . . . . 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 481 | . . . . . 6 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))) |
14 | 13 | ralrimivw 3144 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))) |
15 | 0elpw 5347 | . . . . . . . 8 ⊢ ∅ ∈ 𝒫 No | |
16 | nulssgt 27682 | . . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
17 | 15, 16 | ax-mp 5 | . . . . . . 7 ⊢ ∅ <<s ∅ |
18 | eqscut2 27690 | . . . . . . 7 ⊢ ((∅ <<s ∅ ∧ 𝑋 ∈ No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))))) | |
19 | 17, 18 | mpan 687 | . . . . . 6 ⊢ (𝑋 ∈ No → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥))))) |
20 | 19 | adantr 480 | . . . . 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 2779 | . . 3 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → 𝑋 = 0s ) |
23 | 22 | ex 412 | . 2 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ → 𝑋 = 0s )) |
24 | fveq2 6884 | . . 3 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ( bday ‘ 0s )) | |
25 | bday0s 27712 | . . 3 ⊢ ( bday ‘ 0s ) = ∅ | |
26 | 24, 25 | eqtrdi 2782 | . 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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 ∅c0 4317 𝒫 cpw 4597 {csn 4623 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 No csur 27524 bday cbday 27526 <<s csslt 27664 |s cscut 27666 0s c0s 27706 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 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 6360 df-on 6361 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1o 8464 df-2o 8465 df-no 27527 df-slt 27528 df-bday 27529 df-sslt 27665 df-scut 27667 df-0s 27708 |
This theorem is referenced by: bday1s 27715 cuteq1 27717 0elold 27786 |
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