Theorem nobdaymin 27946

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 6073	. . . 4 ⊢ ( bday “ 𝐴) ⊆ ran bday
2		bdayrn 27944	. . . 4 ⊢ ran bday = On
3	1, 2	sseqtri 3985	. . 3 ⊢ ( bday “ 𝐴) ⊆ On
4		n0 4307	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		bdaydm 27942	. . . . . . . . 9 ⊢ dom bday = No
6	5	sseq2i 3966	. . . . . . . 8 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
7		bdayfun 27940	. . . . . . . . 9 ⊢ Fun bday
8		funfvima2 7229	. . . . . . . . 9 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday ) → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
9	7, 8	mpan 702	. . . . . . . 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 4294	. . . . . . 7 ⊢ (( bday ‘𝑥) ∈ ( bday “ 𝐴) → ( bday “ 𝐴) ≠ ∅)
12	10, 11	syl6 36	. . . . . 6 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → ( bday “ 𝐴) ≠ ∅))
13	12	exlimdv 1963	. . . . 5 ⊢ (𝐴 ⊆ No → (∃𝑥 𝑥 ∈ 𝐴 → ( bday “ 𝐴) ≠ ∅))
14	4, 13	biimtrid 245	. . . 4 ⊢ (𝐴 ⊆ No → (𝐴 ≠ ∅ → ( bday “ 𝐴) ≠ ∅))
15	14	imp 411	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ( bday “ 𝐴) ≠ ∅)
16		onint 7785	. . 3 ⊢ ((( bday “ 𝐴) ⊆ On ∧ ( bday “ 𝐴) ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
17	3, 15, 16	sylancr 598	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
18		bdayfn 27941	. . . 4 ⊢ bday Fn No
19		fvelimab 6953	. . . 4 ⊢ (( bday Fn No ∧ 𝐴 ⊆ No ) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴)))
20	18, 19	mpan 702	. . 3 ⊢ (𝐴 ⊆ No → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴)))
21	20	adantr 485	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   ⊆ wss 3905  ∅c0 4286  ∩ cint 4912  dom cdm 5661  ran crn 5662   “ cima 5664  Oncon0 6360  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536   No csur 27804   bday cbday 27806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1o 8449  df-no 27807  df-bday 27809

This theorem is referenced by:  nocvxmin  27948