| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > elunsn | Structured version Visualization version GIF version | ||
| Description: Elementhood in a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.) |
| Ref | Expression |
|---|---|
| elunsn | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elun 4153 | . 2 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐶})) | |
| 2 | elsng 4640 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶)) | |
| 3 | 2 | orbi2d 916 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶))) |
| 4 | 1, 3 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {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-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 |
| This theorem is referenced by: f1ounsn 7292 chnub 33002 cycpmco2lem7 33152 |
| Copyright terms: Public domain | W3C validator |