Theorem fr0 5609

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 5592	. 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑥((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
2		ss0 4361	. . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
3	2	a1d 25	. . . 4 ⊢ (𝑥 ⊆ ∅ → (¬ ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅ → 𝑥 = ∅))
4	3	necon1ad 2942	. . 3 ⊢ (𝑥 ⊆ ∅ → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
5	4	imp 406	. 2 ⊢ ((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)
6	1, 5	mpgbir 1799	1 ⊢ 𝑅 Fr ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ≠ wne 2925  ∃wrex 3053  {crab 3402   ⊆ wss 3911  ∅c0 4292   class class class wbr 5102   Fr wfr 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-fr 5584

This theorem is referenced by:  we0  5626  frsn  5719  frfi  9208  ifr0  44432