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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 4332 | . 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑠 FrFor 𝑅∅𝑠) | |
2 | 0ss 3461 | . . . 4 ⊢ ∅ ⊆ 𝑠 | |
3 | 2 | a1i 9 | . . 3 ⊢ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → ∅ ⊆ 𝑠) |
4 | df-frfor 4331 | . . 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 ∅ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wral 2455 ⊆ wss 3129 ∅c0 3422 class class class wbr 4003 FrFor wfrfor 4327 Fr wfr 4328 |
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 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 df-frfor 4331 df-frind 4332 |
This theorem is referenced by: we0 4361 |
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