Theorem ufdprmidl 33895

Description: In a unique factorization domain 𝑅, a nonzero prime ideal 𝐽 contains a prime element 𝑝. (Contributed by Thierry Arnoux, 3-Jun-2025.)

Hypotheses

Ref	Expression
isufd.i	⊢ 𝐼 = (PrmIdeal‘𝑅)
isufd.3	⊢ 𝑃 = (RPrime‘𝑅)
isufd.0	⊢ 0 = (0g‘𝑅)
ufdprmidl.2	⊢ (𝜑 → 𝑅 ∈ UFD)
ufdprmidl.3	⊢ (𝜑 → 𝐽 ∈ 𝐼)
ufdprmidl.4	⊢ (𝜑 → 𝐽 ≠ { 0 })

Assertion

Ref	Expression
ufdprmidl	⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)

Distinct variable groups:   𝐽,𝑝   𝑃,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝑅(𝑝)   𝐼(𝑝)   0 (𝑝)

Proof of Theorem ufdprmidl

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 4166	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ 𝑃) = (𝐽 ∩ 𝑃))
2	1	neeq1d 3019	. . . 4 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅))
3	2	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅))
4		incom 4162	. . . . 5 ⊢ (𝑃 ∩ 𝐽) = (𝐽 ∩ 𝑃)
5	4	neeq1i 3024	. . . 4 ⊢ ((𝑃 ∩ 𝐽) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅)
6		inn0 4327	. . . 4 ⊢ ((𝑃 ∩ 𝐽) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)
7	5, 6	bitr3i 280	. . 3 ⊢ ((𝐽 ∩ 𝑃) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)
8	3, 7	bitrdi 290	. 2 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽))
9		ufdprmidl.3	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼)
10		ufdprmidl.4	. . 3 ⊢ (𝜑 → 𝐽 ≠ { 0 })
11	9, 10	eldifsnd 4757	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝐼 ∖ {{ 0 }}))
12		ufdprmidl.2	. . 3 ⊢ (𝜑 → 𝑅 ∈ UFD)
13		isufd.i	. . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅)
14		isufd.3	. . . . 5 ⊢ 𝑃 = (RPrime‘𝑅)
15		isufd.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
16	13, 14, 15	isufd 33894	. . . 4 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅))
17	16	simprbi 503	. . 3 ⊢ (𝑅 ∈ UFD → ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)
18	12, 17	syl 18	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)
19	8, 11, 18	rspcdv2 3578	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∖ cdif 3903   ∩ cin 3905  ∅c0 4286  {csn 4591  ‘cfv 6540  0_gc0g 17514  RPrimecrpm 20560  IDomncidom 20842  PrmIdealcprmidl 21510  UFDcufd 33892

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ufd 33893

This theorem is used by:  1arithufdlem1  33898