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Mirrors > Home > MPE Home > Th. List > Mathboxes > rp-brsslt | Structured version Visualization version GIF version |
Description: Binary relation form of a relation, <, which has been extended from relation 𝑅 to subsets of class 𝑆. Usually, we will assume 𝑅 Or 𝑆. Definition in [Alling], p. 2. Generalization of brsslt 27634. (Originally by Scott Fenton, 8-Dec-2021.) (Contributed by RP, 28-Nov-2023.) |
Ref | Expression |
---|---|
nla0001.defsslt | ⊢ < = {〈𝑎, 𝑏〉 ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} |
Ref | Expression |
---|---|
rp-brsslt | ⊢ (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3999 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝑆 ↔ 𝐴 ⊆ 𝑆)) | |
2 | raleq 3314 | . . 3 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)) | |
3 | 1, 2 | 3anbi13d 1434 | . 2 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦) ↔ (𝐴 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦))) |
4 | sseq1 3999 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝑆 ↔ 𝐵 ⊆ 𝑆)) | |
5 | raleq 3314 | . . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦)) | |
6 | 5 | ralbidv 3169 | . . 3 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦)) |
7 | 4, 6 | 3anbi23d 1435 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐴 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦) ↔ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦))) |
8 | nla0001.defsslt | . 2 ⊢ < = {〈𝑎, 𝑏〉 ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} | |
9 | 3, 7, 8 | bropabg 42562 | 1 ⊢ (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ⊆ wss 3940 class class class wbr 5138 {copab 5200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-xp 5672 |
This theorem is referenced by: nla0002 42664 nla0003 42665 |
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