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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 5534 | . . 3 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 E 𝑥) | |
2 | epel 5471 | . . 3 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥) | |
3 | 1, 2 | sylnib 330 | . 2 ⊢ (( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝑥) |
4 | 3 | ralrimiva 3184 | 1 ⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 class class class wbr 5068 E cep 5466 Fr wfr 5513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-eprel 5467 df-fr 5516 |
This theorem is referenced by: (None) |
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