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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 5555 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
| 3 | opabss 5207 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 | |
| 4 | 2, 3 | sstri 3993 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ 𝑅 |
| 5 | 4 | sseli 3979 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 {copab 5205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-br 5144 df-opab 5206 |
| This theorem is referenced by: opabresex2 7485 fvmptopab 7487 clwlkwlk 29795 |
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