Theorem frminex 5560

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 4316	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑)
2		frminex.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	rabex 5251	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
4		ssrab2 4009	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
5		fri 5540	. . . . . 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 3623	. . . . . . . 8 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
8	7	rexbii 3177	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
9		breq2 5074	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝑥))
10	9	notbid 317	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
11	10	imbi2d 340	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝜓 → ¬ 𝑦𝑅𝑧) ↔ (𝜓 → ¬ 𝑦𝑅𝑥)))
12	11	ralbidv 3120	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
13	12	rexrab2 3632	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
14	8, 13	bitri 274	. . . . . 6 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
15	5, 14	sylib 217	. . . . 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 656	. . . 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 698	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
18	17	ex 412	. 2 ⊢ (𝑅 Fr 𝐴 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))
19	1, 18	syl5bir 242	1 ⊢ (𝑅 Fr 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883  ∅c0 4253   class class class wbr 5070   Fr wfr 5532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-fr 5535

This theorem is referenced by: (None)