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Mirrors > Home > MPE Home > Th. List > eqoreldif | Structured version Visualization version GIF version |
Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.) (Proof shortened by JJ, 23-Jul-2021.) |
Ref | Expression |
---|---|
eqoreldif | ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶) | |
2 | elsni 4608 | . . . . . . 7 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
3 | 2 | con3i 154 | . . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵}) |
4 | 3 | adantl 483 | . . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 ∈ {𝐵}) |
5 | 1, 4 | eldifd 3926 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ (𝐶 ∖ {𝐵})) |
6 | 5 | ex 414 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (¬ 𝐴 = 𝐵 → 𝐴 ∈ (𝐶 ∖ {𝐵}))) |
7 | 6 | orrd 862 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))) |
8 | eleq1a 2833 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐴 ∈ 𝐶)) | |
9 | eldifi 4091 | . . . 4 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴 ∈ 𝐶) | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴 ∈ 𝐶)) |
11 | 8, 10 | jaod 858 | . 2 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})) → 𝐴 ∈ 𝐶)) |
12 | 7, 11 | impbid2 225 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∖ cdif 3912 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-dif 3918 df-sn 4592 |
This theorem is referenced by: lcmfunsnlem2 16523 |
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