Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4983 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)) | |
2 | df-br 4654 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
3 | 2 | biimpi 206 | . . . . 5 ⊢ (𝑥𝑅𝑦 → 〈𝑥, 𝑦〉 ∈ 𝑅) |
4 | eleq1 2689 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
5 | 3, 4 | syl5ibr 236 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝑥𝑅𝑦 → 𝐴 ∈ 𝑅)) |
6 | 5 | imp 445 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅) |
7 | 6 | exlimivv 1860 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅) |
8 | 1, 7 | sylbi 207 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 {copab 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 |
This theorem is referenced by: elopabran 5014 |
Copyright terms: Public domain | W3C validator |