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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 5528 | . . 3 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 E 𝑥) | |
2 | epel 5463 | . . 3 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥) | |
3 | 1, 2 | sylnib 331 | . 2 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝑥) |
4 | 3 | ralrimiva 3105 | 1 ⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2110 ∀wral 3061 class class class wbr 5053 E cep 5459 Fr wfr 5506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-eprel 5460 df-fr 5509 |
This theorem is referenced by: (None) |
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