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| Mirrors > Home > MPE Home > Th. List > ssdifsn | Structured version Visualization version 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 4138 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) → 𝐴 ⊆ 𝐵) | |
| 2 | reldisj 4453 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∩ {𝐶}) = ∅ ↔ 𝐴 ⊆ (𝐵 ∖ {𝐶}))) | |
| 3 | 2 | bicomd 223 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∩ {𝐶}) = ∅)) |
| 4 | 1, 3 | biadanii 822 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅)) |
| 5 | disjsn 4711 | . . 3 ⊢ ((𝐴 ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ 𝐴) | |
| 6 | 5 | anbi2i 623 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
| 7 | 4, 6 | bitri 275 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 |
| This theorem is referenced by: naddcllem 8714 isdomn6 20714 imadrhmcl 20798 isdrng4 33298 drngmxidl 33505 assafld 33688 logdivsqrle 34665 elsetrecslem 49218 |
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