Theorem fr0 4386

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 4367	. 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑠 FrFor 𝑅∅𝑠)
2		0ss 3489	. . . 4 ⊢ ∅ ⊆ 𝑠
3	2	a1i 9	. . 3 ⊢ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → ∅ ⊆ 𝑠)
4		df-frfor 4366	. . 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 1467	1 ⊢ 𝑅 Fr ∅

Syntax hints:   → wi 4  ∀wral 2475   ⊆ wss 3157  ∅c0 3450   class class class wbr 4033   FrFor wfrfor 4362   Fr wfr 4363

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-frfor 4366  df-frind 4367

This theorem is referenced by:  we0  4396