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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 2965 | . 2 ⊢ 𝐴 ⊆ V | |
2 | reldisj 3271 | . 2 ⊢ (𝐴 ⊆ V → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 Vcvv 2557 ∖ cdif 2914 ∩ cin 2916 ⊆ wss 2917 ∅c0 3224 |
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-ral 2311 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 df-nul 3225 |
This theorem is referenced by: ssindif0im 3281 intirr 4711 |
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