Theorem bday0b 28256

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 28250	. . . 4 ⊢ 0s = (∅ |s ∅)
2		snelpwi 5419	. . . . . . 7 ⊢ (𝑋 ∈ No → {𝑋} ∈ 𝒫 No)
3		nulslts 28218	. . . . . . 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 28219	. . . . . . 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 4353	. . . . . . . . 9 ⊢ ∅ ⊆ (bday‘𝑥)
11	9, 10	eqsstrdi 3957	. . . . . . . 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 3162	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (bday‘𝑋) = ∅) → ∀𝑥 ∈ No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → (bday‘𝑋) ⊆ (bday‘𝑥)))
15		0elpw 5324	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 No
16		nulsgts 28219	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
18		eqcuts2 28229	. . . . . . 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 6916	. . 3 ⊢ (𝑋 = 0s → (bday‘𝑋) = (bday‘ 0s ))
25		bday0 28254	. . 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 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {csn 4588   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Nocsur 28054  bdaycbday 28056   <<s cslts 28200   |s ccuts 28202   0_s c0s 28248

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-0s 28250

This theorem is used by:  bday1  28257  cuteq1  28260  0elold  28353  bdayfinbndlem2  28911