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Mirrors > Home > ILE Home > Th. List > prss | 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, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
prss.1 | ⊢ 𝐴 ∈ V |
prss.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
prss | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unss 3220 | . 2 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶) | |
2 | prss.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 2 | snss 3619 | . . 3 ⊢ (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶) |
4 | prss.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | 4 | snss 3619 | . . 3 ⊢ (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶) |
6 | 3, 5 | anbi12i 455 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)) |
7 | df-pr 3504 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
8 | 7 | sseq1i 3093 | . 2 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶) |
9 | 1, 6, 8 | 3bitr4i 211 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1465 Vcvv 2660 ∪ cun 3039 ⊆ wss 3041 {csn 3497 {cpr 3498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 |
This theorem is referenced by: tpss 3655 prsspw 3662 exmidpw 6770 |
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