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Mirrors > Home > MPE Home > Th. List > Mathboxes > brabidga | Structured version Visualization version GIF version |
Description: The law of concretion for a binary relation. Special case of brabga 5527. Usage of this theorem is discouraged because it depends on ax-13 2365, see brabidgaw 37746 for a weaker version that does not require it. (Contributed by Peter Mazsa, 24-Nov-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
brabidga.1 | ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
Ref | Expression |
---|---|
brabidga | ⊢ (𝑥𝑅𝑦 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brabidga.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 | opabid 5518 | . 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 205 = wceq 1533 ∈ wcel 2098 ⟨cop 4629 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 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-13 2365 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 |
This theorem is referenced by: (None) |
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