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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 5527. Version of brabidga 37038 with a disjoint variable condition, which does not require ax-13 2370. (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 5147 | . 2 ⊢ (𝑥𝑅𝑦 ↔ 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
3 | df-br 5142 | . 2 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
4 | opabidw 5517 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
5 | 2, 3, 4 | 3bitri 296 | 1 ⊢ (𝑥𝑅𝑦 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 〈cop 4628 class class class wbr 5141 {copab 5203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 |
This theorem is referenced by: inxpxrn 37068 |
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