Theorem bday0b 27875

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 27869	. . . 4 ⊢ 0s = (∅ |s ∅)
2		snelpwi 5448	. . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No )
3		nulsslt 27842	. . . . . . 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 27843	. . . . . . 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 4400	. . . . . . . . 9 ⊢ ∅ ⊆ ( bday ‘𝑥)
11	9, 10	eqsstrdi 4028	. . . . . . . 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 3150	. . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥)))
15		0elpw 5356	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 No
16		nulssgt 27843	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
18		eqscut2 27851	. . . . . . 7 ⊢ ((∅ <<s ∅ ∧ 𝑋 ∈ No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥)))))
19	17, 18	mpan 690	. . . . . 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 1343	. . . 4 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → (∅ |s ∅) = 𝑋)
22	1, 21	eqtr2id 2790	. . 3 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → 𝑋 = 0s )
23	22	ex 412	. 2 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ → 𝑋 = 0s ))
24		fveq2 6906	. . 3 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ( bday ‘ 0s ))
25		bday0s 27873	. . 3 ⊢ ( bday ‘ 0s ) = ∅
26	24, 25	eqtrdi 2793	. 2 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ∅)
27	23, 26	impbid1 225	1 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431   No csur 27684   bday cbday 27686   <<s csslt 27825   |s cscut 27827   0_s c0s 27867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869

This theorem is referenced by:  bday1s  27876  cuteq1  27878  0elold  27947