Theorem frminex 5604

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 4342	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑)
2		frminex.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	rabex 5285	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
4		ssrab2 4033	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
5		fri 5583	. . . . . 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 3653	. . . . . . . 8 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
8	7	rexbii 3084	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
9		breq2 5103	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝑥))
10	9	notbid 318	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
11	10	imbi2d 340	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝜓 → ¬ 𝑦𝑅𝑧) ↔ (𝜓 → ¬ 𝑦𝑅𝑥)))
12	11	ralbidv 3160	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
13	12	rexrab2 3659	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
14	8, 13	bitri 275	. . . . . 6 ⊢ (∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
15	5, 14	sylib 218	. . . . 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 661	. . . 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 703	. . 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	biimtrrid 243	1 ⊢ (𝑅 Fr 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3400  Vcvv 3441   ⊆ wss 3902  ∅c0 4286   class class class wbr 5099   Fr wfr 5575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-fr 5578

This theorem is referenced by: (None)