Theorem frc 5652

Description: Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 19-Nov-2014.)

Hypothesis

Ref	Expression
frc.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
frc	⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅)

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

Proof of Theorem frc

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frc.1	. . . 4 ⊢ 𝐵 ∈ V
2		fri 5646	. . . 4 ⊢ (((𝐵 ∈ V ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥)
3	1, 2	mpanl1 700	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥)
4	3	3impb 1114	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥)
5		breq1 5151	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
6	5	rabeq0w 4393	. . 3 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥)
7	6	rexbii 3092	. 2 ⊢ (∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∃𝑥 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥)
8	4, 7	sylibr 234	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478   ⊆ wss 3963  ∅c0 4339   class class class wbr 5148   Fr wfr 5638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-fr 5641

This theorem is referenced by:  frirr  5665  epfrc  5674