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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elunsn | Structured version Visualization version GIF version |
Description: Elementhood to 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 4163 | . 2 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐶})) | |
2 | elsng 4645 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶)) | |
3 | 2 | orbi2d 914 | . 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 846 = wceq 1535 ∈ wcel 2104 ∪ cun 3961 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1538 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-v 3479 df-un 3968 df-sn 4632 |
This theorem is referenced by: chnub 32962 cycpmco2lem7 33103 |
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