Theorem nobdaymin 27749

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 6030	. . . 4 ⊢ ( bday “ 𝐴) ⊆ ran bday
2		bdayrn 27747	. . . 4 ⊢ ran bday = On
3	1, 2	sseqtri 3982	. . 3 ⊢ ( bday “ 𝐴) ⊆ On
4		n0 4305	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		bdaydm 27746	. . . . . . . . 9 ⊢ dom bday = No
6	5	sseq2i 3963	. . . . . . . 8 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
7		bdayfun 27744	. . . . . . . . 9 ⊢ Fun bday
8		funfvima2 7177	. . . . . . . . 9 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday ) → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
9	7, 8	mpan 690	. . . . . . . 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 4293	. . . . . . 7 ⊢ (( bday ‘𝑥) ∈ ( bday “ 𝐴) → ( bday “ 𝐴) ≠ ∅)
12	10, 11	syl6 35	. . . . . 6 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → ( bday “ 𝐴) ≠ ∅))
13	12	exlimdv 1934	. . . . 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 7735	. . 3 ⊢ ((( bday “ 𝐴) ⊆ On ∧ ( bday “ 𝐴) ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
17	3, 15, 16	sylancr 587	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
18		bdayfn 27745	. . . 4 ⊢ bday Fn No
19		fvelimab 6906	. . . 4 ⊢ (( bday Fn No ∧ 𝐴 ⊆ No ) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴)))
20	18, 19	mpan 690	. . 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 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060   ⊆ wss 3901  ∅c0 4285  ∩ cint 4902  dom cdm 5624  ran crn 5625   “ cima 5627  Oncon0 6317  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492   No csur 27607   bday cbday 27609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1o 8397  df-no 27610  df-bday 27612

This theorem is referenced by:  nocvxmin  27751