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Mirrors > Home > ILE Home > Th. List > eldifi | GIF version |
Description: Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.) |
Ref | Expression |
---|---|
eldifi | ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3080 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
2 | 1 | simplbi 272 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1480 ∖ cdif 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 |
This theorem is referenced by: difss 3202 ssddif 3310 noel 3367 phpm 6759 fidifsnen 6764 elfi2 6860 fiuni 6866 fifo 6868 fzdifsuc 9861 modfzo0difsn 10168 fsum3cvg 11147 summodclem2a 11150 fisumss 11161 fsumlessfi 11229 binomlem 11252 fproddccvg 11341 prodmodclem2a 11345 oddprmge3 11815 |
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