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Mirrors > Home > ILE Home > Th. List > ssdifsn | GIF version |
Description: Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) (Proof shortened by JJ, 31-May-2022.) |
Ref | Expression |
---|---|
ssdifsn | ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss2 3168 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) → 𝐴 ⊆ 𝐵) | |
2 | reldisj 3378 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∩ {𝐶}) = ∅ ↔ 𝐴 ⊆ (𝐵 ∖ {𝐶}))) | |
3 | 2 | bicomd 140 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∩ {𝐶}) = ∅)) |
4 | 1, 3 | biadan2 449 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅)) |
5 | disjsn 3549 | . . 3 ⊢ ((𝐴 ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ 𝐴) | |
6 | 5 | anbi2i 450 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
7 | 4, 6 | bitri 183 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 = wceq 1312 ∈ wcel 1461 ∖ cdif 3032 ∩ cin 3034 ⊆ wss 3035 ∅c0 3327 {csn 3491 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-v 2657 df-dif 3037 df-in 3041 df-ss 3048 df-nul 3328 df-sn 3497 |
This theorem is referenced by: (None) |
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