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Mirrors > Home > MPE Home > Th. List > elopabr | Structured version Visualization version GIF version |
Description: Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.) |
Ref | Expression |
---|---|
elopabr | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabss 5138 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 | |
2 | 1 | sseli 3917 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5074 {copab 5136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-in 3894 df-ss 3904 df-br 5075 df-opab 5137 |
This theorem is referenced by: (None) |
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