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Mirrors > Home > ILE Home > Th. List > fr0 | GIF version |
Description: Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.) |
Ref | Expression |
---|---|
fr0 | ⊢ 𝑅 Fr ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-frind 4224 | . 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑠 FrFor 𝑅∅𝑠) | |
2 | 0ss 3371 | . . . 4 ⊢ ∅ ⊆ 𝑠 | |
3 | 2 | a1i 9 | . . 3 ⊢ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → ∅ ⊆ 𝑠) |
4 | df-frfor 4223 | . . 3 ⊢ ( FrFor 𝑅∅𝑠 ↔ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → ∅ ⊆ 𝑠)) | |
5 | 3, 4 | mpbir 145 | . 2 ⊢ FrFor 𝑅∅𝑠 |
6 | 1, 5 | mpgbir 1414 | 1 ⊢ 𝑅 Fr ∅ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wral 2393 ⊆ wss 3041 ∅c0 3333 class class class wbr 3899 FrFor wfrfor 4219 Fr wfr 4220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-in 3047 df-ss 3054 df-nul 3334 df-frfor 4223 df-frind 4224 |
This theorem is referenced by: we0 4253 |
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