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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 4570 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
2 | 1 | breqi 5072 | . . 3 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}𝑌 ↔ 𝑋({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉})𝑌) |
3 | brun 5117 | . . 3 ⊢ (𝑋({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉})𝑌 ↔ (𝑋{〈𝐴, 𝐵〉}𝑌 ∨ 𝑋{〈𝐶, 𝐷〉}𝑌)) | |
4 | 2, 3 | bitri 277 | . 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 30429 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))) | |
8 | 5, 6, 7 | syl2anc 586 | . . 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 30429 | . . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (𝑋{〈𝐶, 𝐷〉}𝑌 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))) | |
12 | 9, 10, 11 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑋{〈𝐶, 𝐷〉}𝑌 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))) |
13 | 8, 12 | orbi12d 915 | . 2 ⊢ (𝜑 → ((𝑋{〈𝐴, 𝐵〉}𝑌 ∨ 𝑋{〈𝐶, 𝐷〉}𝑌) ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))) |
14 | 4, 13 | syl5bb 285 | 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 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 {csn 4567 {cpr 4569 〈cop 4573 class class class wbr 5066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 |
This theorem is referenced by: (None) |
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