Theorem fr0 5017

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 5003	. 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑥((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
2		ss0 3926	. . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
3	2	a1d 25	. . . 4 ⊢ (𝑥 ⊆ ∅ → (¬ ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅ → 𝑥 = ∅))
4	3	necon1ad 2799	. . 3 ⊢ (𝑥 ⊆ ∅ → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
5	4	imp 444	. 2 ⊢ ((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)
6	1, 5	mpgbir 1717	1 ⊢ 𝑅 Fr ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ≠ wne 2780  ∃wrex 2897  {crab 2900   ⊆ 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

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-in 3547  df-ss 3554  df-nul 3875  df-fr 4997

This theorem is referenced by:  we0  5033  frsn  5112  frfi  8090  ifr0  37675