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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 | elex 3512 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | 1 | biantrurd 535 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≠ 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵))) |
3 | eldifsn 4719 | . 2 ⊢ (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵)) | |
4 | 2, 3 | syl6rbbr 292 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∖ cdif 3933 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-dif 3939 df-sn 4568 |
This theorem is referenced by: cnvimadfsn 7839 |
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