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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 5031 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}) | |
2 | opex 5321 | . . 3 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
3 | 2 | eltp 4586 | . 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 5333 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)) |
7 | 4, 5 | opth 5333 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐶, 𝐷〉 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)) |
8 | 4, 5 | opth 5333 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐸, 𝐹〉 ↔ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)) |
9 | 6, 7, 8 | 3orbi123i 1153 | . 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 300 | 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 209 ∧ wa 399 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {ctp 4529 〈cop 4531 class class class wbr 5030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-br 5031 |
This theorem is referenced by: sltval2 33276 sltintdifex 33281 sltres 33282 noextendlt 33289 noextendgt 33290 nolesgn2o 33291 sltsolem1 33293 nosepnelem 33297 nosep1o 33299 nosepdmlem 33300 nodenselem8 33308 nodense 33309 nolt02o 33312 nosupbnd2lem1 33328 |
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