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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 5311 | . 2 ⊢ (𝑅 Fr ∅ ↔ ∀𝑥((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) | |
2 | ss0 4201 | . . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
3 | 2 | a1d 25 | . . . 4 ⊢ (𝑥 ⊆ ∅ → (¬ ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅ → 𝑥 = ∅)) |
4 | 3 | necon1ad 3016 | . . 3 ⊢ (𝑥 ⊆ ∅ → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) |
5 | 4 | imp 397 | . 2 ⊢ ((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅) |
6 | 1, 5 | mpgbir 1898 | 1 ⊢ 𝑅 Fr ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ≠ wne 2999 ∃wrex 3118 {crab 3121 ⊆ wss 3798 ∅c0 4146 class class class wbr 4875 Fr wfr 5302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-in 3805 df-ss 3812 df-nul 4147 df-fr 5305 |
This theorem is referenced by: we0 5341 frsn 5428 frfi 8480 ifr0 39487 |
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