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Mirrors > Home > MPE Home > Th. List > elopabr | Structured version Visualization version GIF version |
Description: Membership in a class abstraction of pairs, defined by a binary relation. (Contributed by AV, 16-Feb-2021.) |
Ref | Expression |
---|---|
elopabr | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 5416 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)) | |
2 | df-br 5069 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
3 | 2 | biimpi 218 | . . . . 5 ⊢ (𝑥𝑅𝑦 → 〈𝑥, 𝑦〉 ∈ 𝑅) |
4 | eleq1 2902 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
5 | 3, 4 | syl5ibr 248 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝑥𝑅𝑦 → 𝐴 ∈ 𝑅)) |
6 | 5 | imp 409 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅) |
7 | 6 | exlimivv 1933 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅) |
8 | 1, 7 | sylbi 219 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 {copab 5130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 |
This theorem is referenced by: elopabran 5450 |
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