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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 27185 | . . . 4 ⊢ 0s = (∅ |s ∅) | |
2 | snelpwi 5401 | . . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No ) | |
3 | nulsslt 27158 | . . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋}) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ No → ∅ <<s {𝑋}) |
5 | 4 | adantr 482 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∅ <<s {𝑋}) |
6 | nulssgt 27159 | . . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅) | |
7 | 2, 6 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} <<s ∅) |
8 | 7 | adantr 482 | . . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → {𝑋} <<s ∅) |
9 | id 22 | . . . . . . . . 9 ⊢ (( bday ‘𝑋) = ∅ → ( bday ‘𝑋) = ∅) | |
10 | 0ss 4357 | . . . . . . . . 9 ⊢ ∅ ⊆ ( bday ‘𝑥) | |
11 | 9, 10 | eqsstrdi 3999 | . . . . . . . 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 483 | . . . . . 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 5312 | . . . . . . . 8 ⊢ ∅ ∈ 𝒫 No | |
16 | nulssgt 27159 | . . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
17 | 15, 16 | ax-mp 5 | . . . . . . 7 ⊢ ∅ <<s ∅ |
18 | eqscut2 27167 | . . . . . . 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 482 | . . . . 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 1343 | . . . 4 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → (∅ |s ∅) = 𝑋) |
22 | 1, 21 | eqtr2id 2786 | . . 3 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → 𝑋 = 0s ) |
23 | 22 | ex 414 | . 2 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ → 𝑋 = 0s )) |
24 | fveq2 6843 | . . 3 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ( bday ‘ 0s )) | |
25 | bday0s 27189 | . . 3 ⊢ ( bday ‘ 0s ) = ∅ | |
26 | 24, 25 | eqtrdi 2789 | . 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ⊆ wss 3911 ∅c0 4283 𝒫 cpw 4561 {csn 4587 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 No csur 27004 bday cbday 27006 <<s csslt 27142 |s cscut 27144 0s c0s 27183 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1o 8413 df-2o 8414 df-no 27007 df-slt 27008 df-bday 27009 df-sslt 27143 df-scut 27145 df-0s 27185 |
This theorem is referenced by: bday1s 27192 |
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