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Mirrors > Home > ILE Home > Th. List > disj2 | GIF version |
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.) |
Ref | Expression |
---|---|
disj2 | ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3089 | . 2 ⊢ 𝐴 ⊆ V | |
2 | reldisj 3384 | . 2 ⊢ (𝐴 ⊆ V → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 Vcvv 2660 ∖ cdif 3038 ∩ cin 3040 ⊆ wss 3041 ∅c0 3333 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-dif 3043 df-in 3047 df-ss 3054 df-nul 3334 |
This theorem is referenced by: ssindif0im 3392 intirr 4895 setsresg 11924 setscom 11926 |
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