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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 4791 | . 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) | |
2 | prid1g 4690 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝑥}) | |
3 | eleq2 2901 | . . . 4 ⊢ ({𝐵, 𝐶} = {𝐴, 𝑥} → (𝐴 ∈ {𝐵, 𝐶} ↔ 𝐴 ∈ {𝐴, 𝑥})) | |
4 | 2, 3 | syl5ibrcom 249 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐵, 𝐶} = {𝐴, 𝑥} → 𝐴 ∈ {𝐵, 𝐶})) |
5 | 4 | exlimdv 1930 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} → 𝐴 ∈ {𝐵, 𝐶})) |
6 | 1, 5 | impbid2 228 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cpr 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-dif 3939 df-un 3941 df-nul 4292 df-sn 4562 df-pr 4564 |
This theorem is referenced by: (None) |
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