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Mirrors > Home > MPE Home > Th. List > eldifvsn | Structured version Visualization version GIF version |
Description: A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.) |
Ref | Expression |
---|---|
eldifvsn | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4791 | . 2 ⊢ (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵)) | |
2 | elex 3499 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | 2 | biantrurd 532 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≠ 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵))) |
4 | 1, 3 | bitr4id 290 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 {csn 4631 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-sn 4632 |
This theorem is referenced by: cnvimadfsn 8196 |
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