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Mirrors > Home > MPE Home > Th. List > Mathboxes > brprop | Structured version Visualization version GIF version |
Description: Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
Ref | Expression |
---|---|
brprop.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
brprop.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
brprop.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
brprop.d | ⊢ (𝜑 → 𝐷 ∈ 𝑊) |
Ref | Expression |
---|---|
brprop | ⊢ (𝜑 → (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4519 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
2 | 1 | breqi 5036 | . . 3 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}𝑌 ↔ 𝑋({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉})𝑌) |
3 | brun 5081 | . . 3 ⊢ (𝑋({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉})𝑌 ↔ (𝑋{〈𝐴, 𝐵〉}𝑌 ∨ 𝑋{〈𝐶, 𝐷〉}𝑌)) | |
4 | 2, 3 | bitri 278 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}𝑌 ↔ (𝑋{〈𝐴, 𝐵〉}𝑌 ∨ 𝑋{〈𝐶, 𝐷〉}𝑌)) |
5 | brprop.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | brprop.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | brsnop 5379 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))) | |
8 | 5, 6, 7 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))) |
9 | brprop.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
10 | brprop.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊) | |
11 | brsnop 5379 | . . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (𝑋{〈𝐶, 𝐷〉}𝑌 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))) | |
12 | 9, 10, 11 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑋{〈𝐶, 𝐷〉}𝑌 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))) |
13 | 8, 12 | orbi12d 918 | . 2 ⊢ (𝜑 → ((𝑋{〈𝐴, 𝐵〉}𝑌 ∨ 𝑋{〈𝐶, 𝐷〉}𝑌) ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))) |
14 | 4, 13 | syl5bb 286 | 1 ⊢ (𝜑 → (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ∪ cun 3841 {csn 4516 {cpr 4518 〈cop 4522 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 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-dif 3846 df-un 3848 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 |
This theorem is referenced by: (None) |
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