Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3120 | . 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 2135 ∖ cdif 3108 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 |
This theorem is referenced by: difss 3243 ssddif 3351 noel 3408 phpm 6822 fidifsnen 6827 elfi2 6928 fiuni 6934 fifo 6936 fzdifsuc 10006 modfzo0difsn 10320 fsum3cvg 11305 summodclem2a 11308 fisumss 11319 fsumlessfi 11387 binomlem 11410 fproddccvg 11499 prodmodclem2a 11503 fprodssdc 11517 fprodeq0g 11565 fprodmodd 11568 oddprmge3 12046 oddprm 12170 nnoddn2prm 12171 nnoddn2prmb 12173 2irrexpqap 13443 |
Copyright terms: Public domain | W3C validator |