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Mirrors > Home > MPE Home > Th. List > elopabran | Structured version Visualization version GIF version |
Description: Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.) |
Ref | Expression |
---|---|
elopabran | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝜓) → 𝑥𝑅𝑦) | |
2 | 1 | ssopab2i 5559 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
3 | opabss 5211 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 | |
4 | 2, 3 | sstri 4004 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ 𝑅 |
5 | 4 | sseli 3990 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 class class class wbr 5147 {copab 5209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ss 3979 df-br 5148 df-opab 5210 |
This theorem is referenced by: opabresex2 7484 fvmptopab 7486 clwlkwlk 29807 |
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