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Mirrors > Home > MPE Home > Th. List > brintclab | Structured version Visualization version GIF version |
Description: Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.) |
Ref | Expression |
---|---|
brintclab | ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ∀𝑥(𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5105 | . 2 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ∩ {𝑥 ∣ 𝜑}) | |
2 | opex 5420 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | elintab 4918 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑥)) |
4 | 1, 3 | bitri 275 | 1 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ∀𝑥(𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∈ wcel 2107 {cab 2715 〈cop 4591 ∩ cint 4906 class class class wbr 5104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-int 4907 df-br 5105 |
This theorem is referenced by: brtrclfv 14847 |
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