Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > brtp | Structured version Visualization version GIF version |
Description: A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.) |
Ref | Expression |
---|---|
brtp.1 | ⊢ 𝑋 ∈ V |
brtp.2 | ⊢ 𝑌 ∈ V |
Ref | Expression |
---|---|
brtp | ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5058 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}) | |
2 | opex 5347 | . . 3 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
3 | 2 | eltp 4618 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} ↔ (〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉 ∨ 〈𝑋, 𝑌〉 = 〈𝐶, 𝐷〉 ∨ 〈𝑋, 𝑌〉 = 〈𝐸, 𝐹〉)) |
4 | brtp.1 | . . . 4 ⊢ 𝑋 ∈ V | |
5 | brtp.2 | . . . 4 ⊢ 𝑌 ∈ V | |
6 | 4, 5 | opth 5359 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)) |
7 | 4, 5 | opth 5359 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐶, 𝐷〉 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)) |
8 | 4, 5 | opth 5359 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐸, 𝐹〉 ↔ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)) |
9 | 6, 7, 8 | 3orbi123i 1148 | . 2 ⊢ ((〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉 ∨ 〈𝑋, 𝑌〉 = 〈𝐶, 𝐷〉 ∨ 〈𝑋, 𝑌〉 = 〈𝐸, 𝐹〉) ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹))) |
10 | 1, 3, 9 | 3bitri 298 | 1 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {ctp 4561 〈cop 4563 class class class wbr 5057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-br 5058 |
This theorem is referenced by: sltval2 33060 sltintdifex 33065 sltres 33066 noextendlt 33073 noextendgt 33074 nolesgn2o 33075 sltsolem1 33077 nosepnelem 33081 nosep1o 33083 nosepdmlem 33084 nodenselem8 33092 nodense 33093 nolt02o 33096 nosupbnd2lem1 33112 |
Copyright terms: Public domain | W3C validator |