Theorem nobdaymin 27762

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 6031	. . . 4 ⊢ ( bday “ 𝐴) ⊆ ran bday
2		bdayrn 27760	. . . 4 ⊢ ran bday = On
3	1, 2	sseqtri 3971	. . 3 ⊢ ( bday “ 𝐴) ⊆ On
4		n0 4294	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		bdaydm 27759	. . . . . . . . 9 ⊢ dom bday = No
6	5	sseq2i 3952	. . . . . . . 8 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
7		bdayfun 27757	. . . . . . . . 9 ⊢ Fun bday
8		funfvima2 7180	. . . . . . . . 9 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday ) → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
9	7, 8	mpan 691	. . . . . . . 8 ⊢ (𝐴 ⊆ dom bday → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
10	6, 9	sylbir 235	. . . . . . 7 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
11		ne0i 4282	. . . . . . 7 ⊢ (( bday ‘𝑥) ∈ ( bday “ 𝐴) → ( bday “ 𝐴) ≠ ∅)
12	10, 11	syl6 35	. . . . . 6 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → ( bday “ 𝐴) ≠ ∅))
13	12	exlimdv 1935	. . . . 5 ⊢ (𝐴 ⊆ No → (∃𝑥 𝑥 ∈ 𝐴 → ( bday “ 𝐴) ≠ ∅))
14	4, 13	biimtrid 242	. . . 4 ⊢ (𝐴 ⊆ No → (𝐴 ≠ ∅ → ( bday “ 𝐴) ≠ ∅))
15	14	imp 406	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ( bday “ 𝐴) ≠ ∅)
16		onint 7738	. . 3 ⊢ ((( bday “ 𝐴) ⊆ On ∧ ( bday “ 𝐴) ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
17	3, 15, 16	sylancr 588	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
18		bdayfn 27758	. . . 4 ⊢ bday Fn No
19		fvelimab 6907	. . . 4 ⊢ (( bday Fn No ∧ 𝐴 ⊆ No ) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴)))
20	18, 19	mpan 691	. . 3 ⊢ (𝐴 ⊆ No → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴)))
21	20	adantr 480	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴)))
22	17, 21	mpbid 232	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ⊆ wss 3890  ∅c0 4274  ∩ cint 4890  dom cdm 5625  ran crn 5626   “ cima 5628  Oncon0 6318  Fun wfun 6487   Fn wfn 6488  ‘cfv 6493   No csur 27620   bday cbday 27622

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 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1o 8399  df-no 27623  df-bday 27625

This theorem is referenced by:  nocvxmin  27764