Theorem frminex 5018

Description: If an element of a well-founded set satisfies a property 𝜑, then there is a minimal element that satisfies 𝜑. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
frminex.1	⊢ 𝐴 ∈ V
frminex.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
frminex	⊢ (𝑅 Fr 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))

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

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem frminex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabn0 3912	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑)
2		frminex.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	rabex 4740	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
4		ssrab2 3650	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
5		fri 5000	. . . . . 6 ⊢ ((({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅)) → ∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧)
6		frminex.2	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
7	6	ralrab 3335	. . . . . . . 8 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
8	7	rexbii 3023	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
9		breq2 4587	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝑥))
10	9	notbid 307	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
11	10	imbi2d 329	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝜓 → ¬ 𝑦𝑅𝑧) ↔ (𝜓 → ¬ 𝑦𝑅𝑥)))
12	11	ralbidv 2969	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
13	12	rexrab2 3341	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
14	8, 13	bitri 263	. . . . . 6 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
15	5, 14	sylib 207	. . . . 5 ⊢ ((({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅)) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
16	15	an4s 865	. . . 4 ⊢ ((({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴) ∧ (𝑅 Fr 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅)) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
17	3, 4, 16	mpanl12 714	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
18	17	ex 449	. 2 ⊢ (𝑅 Fr 𝐴 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))
19	1, 18	syl5bir 232	1 ⊢ (𝑅 Fr 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   Fr wfr 4994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-fr 4997

This theorem is referenced by: (None)