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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elsn12g | Structured version Visualization version GIF version |
Description: Join of elsng 4598 and elsn2g 4622. (Contributed by BJ, 18-Nov-2023.) |
Ref | Expression |
---|---|
bj-elsn12g | ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsng 4598 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
2 | elsn2g 4622 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
3 | 1, 2 | jaoi 855 | 1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1541 ∈ wcel 2106 {csn 4584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-sn 4585 |
This theorem is referenced by: (None) |
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