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Mirrors > Home > NFE 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 3437 | . 2 ⊢ (({A} ⊆ C ∧ {B} ⊆ C) ↔ ({A} ∪ {B}) ⊆ C) | |
2 | prss.1 | . . . 4 ⊢ A ∈ V | |
3 | 2 | snss 3838 | . . 3 ⊢ (A ∈ C ↔ {A} ⊆ C) |
4 | prss.2 | . . . 4 ⊢ B ∈ V | |
5 | 4 | snss 3838 | . . 3 ⊢ (B ∈ C ↔ {B} ⊆ C) |
6 | 3, 5 | anbi12i 678 | . 2 ⊢ ((A ∈ C ∧ B ∈ C) ↔ ({A} ⊆ C ∧ {B} ⊆ C)) |
7 | df-pr 3742 | . . 3 ⊢ {A, B} = ({A} ∪ {B}) | |
8 | 7 | sseq1i 3295 | . 2 ⊢ ({A, B} ⊆ C ↔ ({A} ∪ {B}) ⊆ C) |
9 | 1, 6, 8 | 3bitr4i 268 | 1 ⊢ ((A ∈ C ∧ B ∈ C) ↔ {A, B} ⊆ C) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∈ wcel 1710 Vcvv 2859 ∪ cun 3207 ⊆ wss 3257 {csn 3737 {cpr 3738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-ss 3259 df-sn 3741 df-pr 3742 |
This theorem is referenced by: tpss 3871 prsspw 3878 uniintsn 3963 |
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