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Mirrors > Home > ILE Home > Th. List > disj4im | GIF version |
Description: A consequence of two classes being disjoint. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 2-Aug-2018.) |
Ref | Expression |
---|---|
disj4im | ⊢ ((𝐴 ∩ 𝐵) = ∅ → ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj3 3272 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) | |
2 | eqcom 2042 | . . 3 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ (𝐴 ∖ 𝐵) = 𝐴) | |
3 | 1, 2 | bitri 173 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∖ 𝐵) = 𝐴) |
4 | dfpss2 3029 | . . . 4 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ ¬ (𝐴 ∖ 𝐵) = 𝐴)) | |
5 | 4 | simprbi 260 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 → ¬ (𝐴 ∖ 𝐵) = 𝐴) |
6 | 5 | con2i 557 | . 2 ⊢ ((𝐴 ∖ 𝐵) = 𝐴 → ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
7 | 3, 6 | sylbi 114 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ∖ cdif 2914 ∩ cin 2916 ⊆ wss 2917 ⊊ wpss 2918 ∅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-ne 2206 df-ral 2311 df-v 2559 df-dif 2920 df-in 2924 df-pss 2933 df-nul 3225 |
This theorem is referenced by: (None) |
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