Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 4563 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
2 | 1 | breqi 5065 | . . 3 ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}𝑌 ↔ 𝑋({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉})𝑌) |
3 | brun 5110 | . . 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 30427 | . . . 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 30427 | . . . 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 1536 ∈ wcel 2113 ∪ cun 3927 {csn 4560 {cpr 4562 〈cop 4566 class class class wbr 5059 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |