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Mirrors > Home > MPE Home > Th. List > fr0 | Structured version Visualization version 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 | dffr2 5630 | . 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑥((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) | |
2 | ss0 4390 | . . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
3 | 2 | a1d 25 | . . . 4 ⊢ (𝑥 ⊆ ∅ → (¬ ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅ → 𝑥 = ∅)) |
4 | 3 | necon1ad 2949 | . . 3 ⊢ (𝑥 ⊆ ∅ → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) |
5 | 4 | imp 406 | . 2 ⊢ ((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅) |
6 | 1, 5 | mpgbir 1793 | 1 ⊢ 𝑅 Fr ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ≠ wne 2932 ∃wrex 3062 {crab 3424 ⊆ wss 3940 ∅c0 4314 class class class wbr 5138 Fr wfr 5618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-fr 5621 |
This theorem is referenced by: we0 5661 frsn 5753 frfi 9284 ifr0 43698 |
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