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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 4109 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) → 𝐴 ⊆ 𝐵) | |
2 | reldisj 4401 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∩ {𝐶}) = ∅ ↔ 𝐴 ⊆ (𝐵 ∖ {𝐶}))) | |
3 | 2 | bicomd 225 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∩ {𝐶}) = ∅)) |
4 | 1, 3 | biadanii 820 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅)) |
5 | disjsn 4640 | . . 3 ⊢ ((𝐴 ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ 𝐴) | |
6 | 5 | anbi2i 624 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
7 | 4, 6 | bitri 277 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4561 |
This theorem is referenced by: logdivsqrle 31916 elsetrecslem 44795 |
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