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Mirrors > Home > ILE Home > Th. List > ssundifim | GIF version |
Description: A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Ref | Expression |
---|---|
ssundifim | ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ∖ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.6r 836 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)) | |
2 | elun 3084 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
3 | 2 | imbi2i 215 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
4 | eldif 2927 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
5 | 4 | imbi1i 227 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)) |
6 | 1, 3, 5 | 3imtr4i 190 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) → (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) |
7 | 6 | alimi 1344 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) → ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) |
8 | dfss2 2934 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶))) | |
9 | dfss2 2934 | . 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) | |
10 | 7, 8, 9 | 3imtr4i 190 | 1 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ∖ 𝐵) ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∨ wo 629 ∀wal 1241 ∈ wcel 1393 ∖ cdif 2914 ∪ cun 2915 ⊆ 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-un 2922 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
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