Theorem fr0 4416

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 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frind 4397	. 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑠 FrFor 𝑅∅𝑠)
2		0ss 3507	. . . 4 ⊢ ∅ ⊆ 𝑠
3	2	a1i 9	. . 3 ⊢ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → ∅ ⊆ 𝑠)
4		df-frfor 4396	. . 3 ⊢ ( FrFor 𝑅∅𝑠 ↔ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → ∅ ⊆ 𝑠))
5	3, 4	mpbir 146	. 2 ⊢ FrFor 𝑅∅𝑠
6	1, 5	mpgbir 1477	1 ⊢ 𝑅 Fr ∅

Syntax hints:   → wi 4  ∀wral 2486   ⊆ wss 3174  ∅c0 3468   class class class wbr 4059   FrFor wfrfor 4392   Fr wfr 4393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469  df-frfor 4396  df-frind 4397

This theorem is referenced by:  we0  4426