Theorem nobdaymin 27745

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 6036	. . . 4 ⊢ ( bday “ 𝐴) ⊆ ran bday
2		bdayrn 27743	. . . 4 ⊢ ran bday = On
3	1, 2	sseqtri 3970	. . 3 ⊢ ( bday “ 𝐴) ⊆ On
4		n0 4293	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		bdaydm 27742	. . . . . . . . 9 ⊢ dom bday = No
6	5	sseq2i 3951	. . . . . . . 8 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
7		bdayfun 27740	. . . . . . . . 9 ⊢ Fun bday
8		funfvima2 7186	. . . . . . . . 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 4281	. . . . . . 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 7744	. . 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 27741	. . . 4 ⊢ bday Fn No
19		fvelimab 6912	. . . 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 2932  ∃wrex 3061   ⊆ wss 3889  ∅c0 4273  ∩ cint 4889  dom cdm 5631  ran crn 5632   “ cima 5634  Oncon0 6323  Fun wfun 6492   Fn wfn 6493  ‘cfv 6498   No csur 27603   bday cbday 27605

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-1o 8405  df-no 27606  df-bday 27608

This theorem is referenced by:  nocvxmin  27747