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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 3075 | . 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 3063 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 |
This theorem is referenced by: difss 3197 ssddif 3305 noel 3362 phpm 6752 fidifsnen 6757 elfi2 6853 fiuni 6859 fifo 6861 fzdifsuc 9854 modfzo0difsn 10161 fsum3cvg 11139 summodclem2a 11143 fisumss 11154 fsumlessfi 11222 binomlem 11245 fproddccvg 11334 oddprmge3 11804 |
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