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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 3066 | . 2 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴) | |
2 | 1 | ssriv 2949 | 1 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 2914 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 |
This theorem is referenced by: difssd 3071 difss2 3072 ssdifss 3074 0dif 3295 undif1ss 3298 undifabs 3300 inundifss 3301 undifss 3303 difsnpssim 3507 unidif 3612 iunxdif2 3705 difexg 3898 reldif 4457 cnvdif 4730 resdif 5148 fndmdif 5272 swoer 6134 swoord1 6135 swoord2 6136 phplem2 6316 phpm 6327 pinn 6407 niex 6410 dmaddpi 6423 dmmulpi 6424 lerelxr 7082 |
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