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| Mirrors > Home > MPE Home > Th. List > brtp | Structured version Visualization version GIF version | ||
| Description: A necessary and sufficient condition for two sets to be related under a binary relation which is 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 5144 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}) | |
| 2 | opex 5469 | . . 3 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
| 3 | 2 | eltp 4689 | . 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 5481 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)) |
| 7 | 4, 5 | opth 5481 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐶, 𝐷〉 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)) |
| 8 | 4, 5 | opth 5481 | . . 3 ⊢ (〈𝑋, 𝑌〉 = 〈𝐸, 𝐹〉 ↔ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)) |
| 9 | 6, 7, 8 | 3orbi123i 1157 | . 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 297 | 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 206 ∧ wa 395 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {ctp 4630 〈cop 4632 class class class wbr 5143 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-br 5144 |
| This theorem is referenced by: sltval2 27701 sltintdifex 27706 sltres 27707 noextendlt 27714 noextendgt 27715 nolesgn2o 27716 nogesgn1o 27718 sltsolem1 27720 nosepnelem 27724 nosep1o 27726 nosep2o 27727 nosepdmlem 27728 nodenselem8 27736 nodense 27737 nolt02o 27740 nogt01o 27741 nosupbnd2lem1 27760 noinfbnd2lem1 27775 |
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