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| Mirrors > Home > MPE Home > Th. List > Mathboxes > efrunt | Structured version Visualization version GIF version | ||
| Description: If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) |
| Ref | Expression |
|---|---|
| efrunt | ⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frirr 5661 | . . 3 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 E 𝑥) | |
| 2 | epel 5587 | . . 3 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥) | |
| 3 | 1, 2 | sylnib 328 | . 2 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝑥) |
| 4 | 3 | ralrimiva 3146 | 1 ⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 E cep 5583 Fr wfr 5634 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-fr 5637 |
| This theorem is referenced by: (None) |
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