Theorem bday0b 27320

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 27314	. . . 4 ⊢ 0s = (∅ |s ∅)
2		snelpwi 5442	. . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No )
3		nulsslt 27287	. . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋})
4	2, 3	syl 17	. . . . . 6 ⊢ (𝑋 ∈ No → ∅ <<s {𝑋})
5	4	adantr 481	. . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∅ <<s {𝑋})
6		nulssgt 27288	. . . . . . 7 ⊢ ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅)
7	2, 6	syl 17	. . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} <<s ∅)
8	7	adantr 481	. . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → {𝑋} <<s ∅)
9		id 22	. . . . . . . . 9 ⊢ (( bday ‘𝑋) = ∅ → ( bday ‘𝑋) = ∅)
10		0ss 4395	. . . . . . . . 9 ⊢ ∅ ⊆ ( bday ‘𝑥)
11	9, 10	eqsstrdi 4035	. . . . . . . 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 482	. . . . . 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 5353	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 No
16		nulssgt 27288	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
18		eqscut2 27296	. . . . . . 7 ⊢ ((∅ <<s ∅ ∧ 𝑋 ∈ No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥)))))
19	17, 18	mpan 688	. . . . . 6 ⊢ (𝑋 ∈ No → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥)))))
20	19	adantr 481	. . . . 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 1342	. . . 4 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → (∅ |s ∅) = 𝑋)
22	1, 21	eqtr2id 2785	. . 3 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → 𝑋 = 0s )
23	22	ex 413	. 2 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ → 𝑋 = 0s ))
24		fveq2 6888	. . 3 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ( bday ‘ 0s ))
25		bday0s 27318	. . 3 ⊢ ( bday ‘ 0s ) = ∅
26	24, 25	eqtrdi 2788	. 2 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ∅)
27	23, 26	impbid1 224	1 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  {csn 4627   class class class wbr 5147  ‘cfv 6540  (class class class)co 7405   No csur 27132   bday cbday 27134   <<s csslt 27271   |s cscut 27273   0_s c0s 27312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1o 8462  df-2o 8463  df-no 27135  df-slt 27136  df-bday 27137  df-sslt 27272  df-scut 27274  df-0s 27314

This theorem is referenced by:  bday1s  27321  cuteq1  27323  0elold  27391