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| Mirrors > Home > MPE Home > Th. List > elpreqprb | Structured version Visualization version GIF version | ||
| Description: A set is an element of an unordered pair iff there is another (maybe the same) set which is an element of the unordered pair. (Proposed by BJ, 8-Dec-2020.) (Contributed by AV, 9-Dec-2020.) | 
| Ref | Expression | 
|---|---|
| elpreqprb | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elpreqpr 4866 | . 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) | |
| 2 | prid1g 4759 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝑥}) | |
| 3 | eleq2 2829 | . . . 4 ⊢ ({𝐵, 𝐶} = {𝐴, 𝑥} → (𝐴 ∈ {𝐵, 𝐶} ↔ 𝐴 ∈ {𝐴, 𝑥})) | |
| 4 | 2, 3 | syl5ibrcom 247 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐵, 𝐶} = {𝐴, 𝑥} → 𝐴 ∈ {𝐵, 𝐶})) | 
| 5 | 4 | exlimdv 1932 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} → 𝐴 ∈ {𝐵, 𝐶})) | 
| 6 | 1, 5 | impbid2 226 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cpr 4627 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-un 3955 df-nul 4333 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: (None) | 
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