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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 5070 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}) | |
2 | opex 5359 | . . 3 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
3 | 2 | eltp 4629 | . 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 5371 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)) |
7 | 4, 5 | opth 5371 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐶, 𝐷〉 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)) |
8 | 4, 5 | opth 5371 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐸, 𝐹〉 ↔ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)) |
9 | 6, 7, 8 | 3orbi123i 1152 | . 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 299 | 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 208 ∧ wa 398 ∨ w3o 1082 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {ctp 4574 〈cop 4576 class class class wbr 5069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-br 5070 |
This theorem is referenced by: sltval2 33167 sltintdifex 33172 sltres 33173 noextendlt 33180 noextendgt 33181 nolesgn2o 33182 sltsolem1 33184 nosepnelem 33188 nosep1o 33190 nosepdmlem 33191 nodenselem8 33199 nodense 33200 nolt02o 33203 nosupbnd2lem1 33219 |
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