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Mirrors > Home > ILE Home > Th. List > prssg | GIF version |
Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
prssg | ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ((A ∈ 𝐶 ∧ B ∈ 𝐶) ↔ {A, B} ⊆ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssg 3491 | . . 3 ⊢ (A ∈ 𝑉 → (A ∈ 𝐶 ↔ {A} ⊆ 𝐶)) | |
2 | snssg 3491 | . . 3 ⊢ (B ∈ 𝑊 → (B ∈ 𝐶 ↔ {B} ⊆ 𝐶)) | |
3 | 1, 2 | bi2anan9 538 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ((A ∈ 𝐶 ∧ B ∈ 𝐶) ↔ ({A} ⊆ 𝐶 ∧ {B} ⊆ 𝐶))) |
4 | unss 3111 | . . 3 ⊢ (({A} ⊆ 𝐶 ∧ {B} ⊆ 𝐶) ↔ ({A} ∪ {B}) ⊆ 𝐶) | |
5 | df-pr 3374 | . . . 4 ⊢ {A, B} = ({A} ∪ {B}) | |
6 | 5 | sseq1i 2963 | . . 3 ⊢ ({A, B} ⊆ 𝐶 ↔ ({A} ∪ {B}) ⊆ 𝐶) |
7 | 4, 6 | bitr4i 176 | . 2 ⊢ (({A} ⊆ 𝐶 ∧ {B} ⊆ 𝐶) ↔ {A, B} ⊆ 𝐶) |
8 | 3, 7 | syl6bb 185 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ((A ∈ 𝐶 ∧ B ∈ 𝐶) ↔ {A, B} ⊆ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 ∪ cun 2909 ⊆ wss 2911 {csn 3367 {cpr 3368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 |
This theorem is referenced by: prssi 3513 prsspwg 3514 |
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