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Theorem bday0b 33584
 Description: The only surreal with birthday ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
Assertion
Ref Expression
bday0b (𝑋 No → (( bday 𝑋) = ∅ ↔ 𝑋 = 0s ))

Proof of Theorem bday0b
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-0s 33578 . . . 4 0s = (∅ |s ∅)
2 snelpwi 5305 . . . . . . 7 (𝑋 No → {𝑋} ∈ 𝒫 No )
3 nulsslt 33554 . . . . . . 7 ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋})
42, 3syl 17 . . . . . 6 (𝑋 No → ∅ <<s {𝑋})
54adantr 484 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ∅ <<s {𝑋})
6 nulssgt 33555 . . . . . . 7 ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅)
72, 6syl 17 . . . . . 6 (𝑋 No → {𝑋} <<s ∅)
87adantr 484 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → {𝑋} <<s ∅)
9 id 22 . . . . . . . . 9 (( bday 𝑋) = ∅ → ( bday 𝑋) = ∅)
10 0ss 4292 . . . . . . . . 9 ∅ ⊆ ( bday 𝑥)
119, 10eqsstrdi 3946 . . . . . . . 8 (( bday 𝑋) = ∅ → ( bday 𝑋) ⊆ ( bday 𝑥))
1211a1d 25 . . . . . . 7 (( bday 𝑋) = ∅ → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
1312adantl 485 . . . . . 6 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
1413ralrimivw 3114 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
15 0elpw 5224 . . . . . . . 8 ∅ ∈ 𝒫 No
16 nulssgt 33555 . . . . . . . 8 (∅ ∈ 𝒫 No → ∅ <<s ∅)
1715, 16ax-mp 5 . . . . . . 7 ∅ <<s ∅
18 eqscut2 33561 . . . . . . 7 ((∅ <<s ∅ ∧ 𝑋 No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
1917, 18mpan 689 . . . . . 6 (𝑋 No → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
2019adantr 484 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
215, 8, 14, 20mpbir3and 1339 . . . 4 ((𝑋 No ∧ ( bday 𝑋) = ∅) → (∅ |s ∅) = 𝑋)
221, 21syl5req 2806 . . 3 ((𝑋 No ∧ ( bday 𝑋) = ∅) → 𝑋 = 0s )
2322ex 416 . 2 (𝑋 No → (( bday 𝑋) = ∅ → 𝑋 = 0s ))
24 fveq2 6658 . . 3 (𝑋 = 0s → ( bday 𝑋) = ( bday ‘ 0s ))
25 bday0s 33582 . . 3 ( bday ‘ 0s ) = ∅
2624, 25eqtrdi 2809 . 2 (𝑋 = 0s → ( bday 𝑋) = ∅)
2723, 26impbid1 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 3858  ∅c0 4225  𝒫 cpw 4494  {csn 4522   class class class wbr 5032  ‘cfv 6335  (class class class)co 7150   No csur 33408   bday cbday 33410   <
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