Theorem fr0 5667

Description: Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)

Assertion

Ref	Expression
fr0	⊢ 𝑅 Fr ∅

Proof of Theorem fr0

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffr2 5650	. 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑥((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
2		ss0 4408	. . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
3	2	a1d 25	. . . 4 ⊢ (𝑥 ⊆ ∅ → (¬ ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅ → 𝑥 = ∅))
4	3	necon1ad 2955	. . 3 ⊢ (𝑥 ⊆ ∅ → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
5	4	imp 406	. 2 ⊢ ((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)
6	1, 5	mpgbir 1796	1 ⊢ 𝑅 Fr ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ≠ wne 2938  ∃wrex 3068  {crab 3433   ⊆ 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-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-fr 5641

This theorem is referenced by:  we0  5684  frsn  5776  frfi  9319  ifr0  44446