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Mirrors > Home > ILE Home > Th. List > difss | GIF version |
Description: Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.) |
Ref | Expression |
---|---|
difss | ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 3193 | . 2 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴) | |
2 | 1 | ssriv 3096 | 1 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 3063 ⊆ wss 3066 |
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 df-in 3072 df-ss 3079 |
This theorem is referenced by: difssd 3198 difss2 3199 ssdifss 3201 0dif 3429 undif1ss 3432 undifabs 3434 inundifss 3435 undifss 3438 unidif 3763 iunxdif2 3856 difexg 4064 reldif 4654 cnvdif 4940 resdif 5382 fndmdif 5518 swoer 6450 swoord1 6451 swoord2 6452 phplem2 6740 phpm 6752 unfiin 6807 sbthlem2 6839 sbthlemi4 6841 sbthlemi5 6842 difinfinf 6979 pinn 7110 niex 7113 dmaddpi 7126 dmmulpi 7127 lerelxr 7820 fisumss 11154 structcnvcnv 11964 strleund 12036 strleun 12037 strle1g 12038 discld 12294 exmid1stab 13184 |
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