Theorem nobdaymin 28196

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 6216	. . . 4 ⊢ (bday “ 𝐴) ⊆ ran bday
2		bdayrn 28194	. . . 4 ⊢ ran bday = On
3	1, 2	sseqtri 3961	. . 3 ⊢ (bday “ 𝐴) ⊆ On
4		n0 4303	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		bdaydm 28192	. . . . . . . . 9 ⊢ dom bday = No
6	5	sseq2i 3942	. . . . . . . 8 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No)
7		bdayfun 28190	. . . . . . . . 9 ⊢ Fun bday
8		funfvima2 7268	. . . . . . . . 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 4290	. . . . . . 7 ⊢ ((bday‘𝑥) ∈ (bday “ 𝐴) → (bday “ 𝐴) ≠ ∅)
12	10, 11	syl6 36	. . . . . 6 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → (bday “ 𝐴) ≠ ∅))
13	12	exlimdv 1968	. . . . 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 7838	. . 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 28191	. . . 4 ⊢ bday Fn No
19		fvelimab 6988	. . . 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 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  ∩ cint 4912  dom cdm 5658  ran crn 5659   “ cima 5661  Oncon0 6388  Fun wfun 6560   Fn wfn 6561  ‘cfv 6566  Nocsur 28054  bdaycbday 28056

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-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-rab 3417  df-v 3456  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-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-fo 6572  df-fv 6574  df-1o 8514  df-no 28057  df-bday 28059

This theorem is used by:  nocvxmin  28198