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Mirrors > Home > MPE Home > Th. List > Mathboxes > brabidgaw | Structured version Visualization version GIF version |
Description: The law of concretion for a binary relation. Special case of brabga 5400. Version of brabidga 36182 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by Gino Giotto, 2-Apr-2024.) |
Ref | Expression |
---|---|
brabidgaw.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
brabidgaw | ⊢ (𝑥𝑅𝑦 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brabidgaw.1 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | 1 | breqi 5045 | . 2 ⊢ (𝑥𝑅𝑦 ↔ 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
3 | df-br 5040 | . 2 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
4 | opabidw 5391 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
5 | 2, 3, 4 | 3bitri 300 | 1 ⊢ (𝑥𝑅𝑦 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2112 〈cop 4533 class class class wbr 5039 {copab 5101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 |
This theorem is referenced by: inxpxrn 36207 |
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