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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 5540. Usage of this theorem is discouraged because it depends on ax-13 2366, see brabidgaw 37869 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 5158 | . 2 ⊢ (𝑥𝑅𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦) |
3 | df-br 5153 | . 2 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) | |
4 | opabid 5531 | . 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 1533 ∈ wcel 2098 ⟨cop 4638 class class class wbr 5152 {copab 5214 |
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 2166 ax-13 2366 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 |
This theorem is referenced by: (None) |
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