Theorem bday0b 28211

Description: The only surreal with birthday ∅ is 0_s. (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.

Step	Hyp	Ref	Expression
1		df-0s 28205	. . . 4 ⊢ 0s = (∅ |s ∅)
2		snelpwi 5415	. . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No)
3		nulslts 28173	. . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋})
4	2, 3	syl 18	. . . . . 6 ⊢ (𝑋 ∈ No → ∅ <<s {𝑋})
5	4	adantr 486	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (bday‘𝑋) = ∅) → ∅ <<s {𝑋})
6		nulsgts 28174	. . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅)
7	2, 6	syl 18	. . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} <<s ∅)
8	7	adantr 486	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (bday‘𝑋) = ∅) → {𝑋} <<s ∅)
9		id 23	. . . . . . . . 9 ⊢ ((bday‘𝑋) = ∅ → (bday‘𝑋) = ∅)
10		0ss 4352	. . . . . . . . 9 ⊢ ∅ ⊆ (bday‘𝑥)
11	9, 10	eqsstrdi 3956	. . . . . . . 8 ⊢ ((bday‘𝑋) = ∅ → (bday‘𝑋) ⊆ (bday‘𝑥))
12	11	a1d 26	. . . . . . 7 ⊢ ((bday‘𝑋) = ∅ → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → (bday‘𝑋) ⊆ (bday‘𝑥)))
13	12	adantl 487	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ (bday‘𝑋) = ∅) → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → (bday‘𝑋) ⊆ (bday‘𝑥)))
14	13	ralrimivw 3161	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (bday‘𝑋) = ∅) → ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → (bday‘𝑋) ⊆ (bday‘𝑥)))
15		0elpw 5320	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 No
16		nulsgts 28174	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
18		eqcuts2 28184	. . . . . . 7 ⊢ ((∅ <<s ∅ ∧ 𝑋 ∈ No) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → (bday‘𝑋) ⊆ (bday‘𝑥)))))
19	17, 18	mpan 703	. . . . . 6 ⊢ (𝑋 ∈ No → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → (bday‘𝑋) ⊆ (bday‘𝑥)))))
20	19	adantr 486	. . . . 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 1361	. . . 4 ⊢ ((𝑋 ∈ No ∧ (bday‘𝑋) = ∅) → (∅ |s ∅) = 𝑋)
22	1, 21	eqtr2id 2811	. . 3 ⊢ ((𝑋 ∈ No ∧ (bday‘𝑋) = ∅) → 𝑋 = 0s )
23	22	ex 418	. 2 ⊢ (𝑋 ∈ No → ((bday‘𝑋) = ∅ → 𝑋 = 0s ))
24		fveq2 6893	. . 3 ⊢ (𝑋 = 0s → (bday‘𝑋) = (bday‘ 0s ))
25		bday0 28209	. . 3 ⊢ (bday‘ 0s ) = ∅
26	24, 25	eqtrdi 2814	. 2 ⊢ (𝑋 = 0s → (bday‘𝑋) = ∅)
27	23, 26	impbid1 228	1 ⊢ (𝑋 ∈ No → ((bday‘𝑋) = ∅ ↔ 𝑋 = 0s ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3079   ⊆ wss 3899  ∅c0 4280  𝒫 cpw 4559  {csn 4587   class class class wbr 5105  ‘cfv 6546  (class class class)co 7427  Nocsur 28009  bdaycbday 28011   <<s cslts 28155   |s ccuts 28157   0_s c0s 28203

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-uni 4870  df-int 4910  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1o 8487  df-2o 8488  df-no 28012  df-lts 28013  df-bday 28014  df-slts 28156  df-cuts 28158  df-0s 28205

This theorem is used by:  bday1  28212  cuteq1  28215  0elold  28308  bdayfinbndlem2  28866