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 4584 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}) | |
2 | opex 4859 | . . 3 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
3 | 2 | eltp 4177 | . 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 4871 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)) |
7 | 4, 5 | opth 4871 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐶, 𝐷〉 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)) |
8 | 4, 5 | opth 4871 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐸, 𝐹〉 ↔ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)) |
9 | 6, 7, 8 | 3orbi123i 1245 | . 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 285 | 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 195 ∧ wa 383 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {ctp 4129 〈cop 4131 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-br 4584 |
This theorem is referenced by: sltval2 31053 sltsgn1 31058 sltsgn2 31059 sltintdifex 31060 sltres 31061 sltsolem1 31067 nodenselem8 31087 nodense 31088 nobndup 31099 nobnddown 31100 |
Copyright terms: Public domain | W3C validator |