Theorem nobdaymin 28148

Description: Any non-empty class of surreals has a birthday-minimal element. (Contributed by Scott Fenton, 11-Dec-2025.)

Assertion

Ref	Expression
nobdaymin	⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (bday‘𝑥) = ∩ (bday “ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem nobdaymin

Step	Hyp	Ref	Expression
1		imassrn 6200	. . . 4 ⊢ (bday “ 𝐴) ⊆ ran bday
2		bdayrn 28146	. . . 4 ⊢ ran bday = On
3	1, 2	sseqtri 3957	. . 3 ⊢ (bday “ 𝐴) ⊆ On
4		n0 4299	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		bdaydm 28144	. . . . . . . . 9 ⊢ dom bday = No
6	5	sseq2i 3938	. . . . . . . 8 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No)
7		bdayfun 28142	. . . . . . . . 9 ⊢ Fun bday
8		funfvima2 7241	. . . . . . . . 9 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday) → (𝑥 ∈ 𝐴 → (bday‘𝑥) ∈ (bday “ 𝐴)))
9	7, 8	mpan 703	. . . . . . . 8 ⊢ (𝐴 ⊆ dom bday → (𝑥 ∈ 𝐴 → (bday‘𝑥) ∈ (bday “ 𝐴)))
10	6, 9	sylbir 238	. . . . . . 7 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → (bday‘𝑥) ∈ (bday “ 𝐴)))
11		ne0i 4286	. . . . . . 7 ⊢ ((bday‘𝑥) ∈ (bday “ 𝐴) → (bday “ 𝐴) ≠ ∅)
12	10, 11	syl6 36	. . . . . 6 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → (bday “ 𝐴) ≠ ∅))
13	12	exlimdv 1965	. . . . 5 ⊢ (𝐴 ⊆ No → (∃𝑥 𝑥 ∈ 𝐴 → (bday “ 𝐴) ≠ ∅))
14	4, 13	biimtrid 245	. . . 4 ⊢ (𝐴 ⊆ No → (𝐴 ≠ ∅ → (bday “ 𝐴) ≠ ∅))
15	14	imp 412	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → (bday “ 𝐴) ≠ ∅)
16		onint 7808	. . 3 ⊢ (((bday “ 𝐴) ⊆ On ∧ (bday “ 𝐴) ≠ ∅) → ∩ (bday “ 𝐴) ∈ (bday “ 𝐴))
17	3, 15, 16	sylancr 599	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∩ (bday “ 𝐴) ∈ (bday “ 𝐴))
18		bdayfn 28143	. . . 4 ⊢ bday Fn No
19		fvelimab 6962	. . . 4 ⊢ ((bday Fn No ∧ 𝐴 ⊆ No) → (∩ (bday “ 𝐴) ∈ (bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (bday‘𝑥) = ∩ (bday “ 𝐴)))
20	18, 19	mpan 703	. . 3 ⊢ (𝐴 ⊆ No → (∩ (bday “ 𝐴) ∈ (bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (bday‘𝑥) = ∩ (bday “ 𝐴)))
21	20	adantr 486	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → (∩ (bday “ 𝐴) ∈ (bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (bday‘𝑥) = ∩ (bday “ 𝐴)))
22	17, 21	mpbid 235	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (bday‘𝑥) = ∩ (bday “ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   ⊆ wss 3896  ∅c0 4277  ∩ cint 4906  dom cdm 5651  ran crn 5652   “ cima 5654  Oncon0 6365  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543  Nocsur 28006  bdaycbday 28008

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

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 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1o 8484  df-no 28009  df-bday 28011

This theorem is used by:  nocvxmin  28150