Theorem fr0 4352

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 4333	. 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑠 FrFor 𝑅∅𝑠)
2		0ss 3462	. . . 4 ⊢ ∅ ⊆ 𝑠
3	2	a1i 9	. . 3 ⊢ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → ∅ ⊆ 𝑠)
4		df-frfor 4332	. . 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 1453	1 ⊢ 𝑅 Fr ∅

Syntax hints:   → wi 4  ∀wral 2455   ⊆ wss 3130  ∅c0 3423   class class class wbr 4004   FrFor wfrfor 4328   Fr wfr 4329

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-nul 3424  df-frfor 4332  df-frind 4333

This theorem is referenced by:  we0  4362