Theorem frd 5615

Description: A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (deduction form). (Contributed by BJ, 16-Nov-2024.)

Hypotheses

Ref	Expression
frd.fr	⊢ (𝜑 → 𝑅 Fr 𝐴)
frd.ss	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
frd.ex	⊢ (𝜑 → 𝐵 ∈ 𝑉)
frd.n0	⊢ (𝜑 → 𝐵 ≠ ∅)

Assertion

Ref	Expression
frd	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem frd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
2		biidd 262	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝐵) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑥))
3	1, 2	raleqbidv 3329	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝐵) → (∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
4	1, 3	rexeqbidv 3330	. 2 ⊢ ((𝜑 ∧ 𝑧 = 𝐵) → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
5		frd.ex	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
6		frd.ss	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
7	5, 6	elpwd 4586	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴)
8		frd.n0	. . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅)
9		nelsn 4647	. . . 4 ⊢ (𝐵 ≠ ∅ → ¬ 𝐵 ∈ {∅})
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ {∅})
11	7, 10	eldifd 3942	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝒫 𝐴 ∖ {∅}))
12		frd.fr	. . 3 ⊢ (𝜑 → 𝑅 Fr 𝐴)
13		dffr6 5614	. . 3 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑧 ∈ (𝒫 𝐴 ∖ {∅})∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥)
14	12, 13	sylib 218	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝒫 𝐴 ∖ {∅})∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥)
15	4, 11, 14	rspcdv2 3601	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ∖ cdif 3928   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606   class class class wbr 5124   Fr wfr 5608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-v 3466  df-dif 3934  df-ss 3948  df-pw 4582  df-sn 4607  df-fr 5611

This theorem is referenced by:  fri  5616  frxp3  8155  weiunfr  36490