Theorem bday0b 27805

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 27799	. . . 4 ⊢ 0s = (∅ |s ∅)
2		snelpwi 5396	. . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No )
3		nulslts 27767	. . . . . . 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		nulsgts 27768	. . . . . . 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 4340	. . . . . . . . 9 ⊢ ∅ ⊆ ( bday ‘𝑥)
11	9, 10	eqsstrdi 3966	. . . . . . . 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 3133	. . . . 5 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥)))
15		0elpw 5297	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 No
16		nulsgts 27768	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
18		eqcuts2 27778	. . . . . . 7 ⊢ ((∅ <<s ∅ ∧ 𝑋 ∈ No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑥)))))
19	17, 18	mpan 691	. . . . . 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 1344	. . . 4 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → (∅ |s ∅) = 𝑋)
22	1, 21	eqtr2id 2784	. . 3 ⊢ ((𝑋 ∈ No ∧ ( bday ‘𝑋) = ∅) → 𝑋 = 0s )
23	22	ex 412	. 2 ⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ → 𝑋 = 0s ))
24		fveq2 6840	. . 3 ⊢ (𝑋 = 0s → ( bday ‘𝑋) = ( bday ‘ 0s ))
25		bday0 27803	. . 3 ⊢ ( bday ‘ 0s ) = ∅
26	24, 25	eqtrdi 2787	. 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 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  {csn 4567   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367   No csur 27603   bday cbday 27605   <<s cslts 27749   |s ccuts 27751   0_s c0s 27797

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-0s 27799

This theorem is referenced by:  bday1  27806  cuteq1  27809  0elold  27902  bdayfinbndlem2  28460