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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 5544. Version of brabidga 38348 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by GG, 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 5154 | . 2 ⊢ (𝑥𝑅𝑦 ↔ 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
3 | df-br 5149 | . 2 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
4 | opabidw 5534 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
5 | 2, 3, 4 | 3bitri 297 | 1 ⊢ (𝑥𝑅𝑦 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 {copab 5210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 |
This theorem is referenced by: inxpxrn 38377 |
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