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Mirrors > Home > NFE Home > Th. List > prex | GIF version |
Description: An unordered pair exists. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
prex | ⊢ {A, B} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3742 | . 2 ⊢ {A, B} = ({A} ∪ {B}) | |
2 | snex 4111 | . . 3 ⊢ {A} ∈ V | |
3 | snex 4111 | . . 3 ⊢ {B} ∈ V | |
4 | 2, 3 | unex 4106 | . 2 ⊢ ({A} ∪ {B}) ∈ V |
5 | 1, 4 | eqeltri 2423 | 1 ⊢ {A, B} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 Vcvv 2859 ∪ cun 3207 {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 ax-nin 4078 ax-sn 4087 |
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-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 |
This theorem is referenced by: opkex 4113 elopk 4129 opkthg 4131 enprmaplem5 6080 2p1e3c 6156 ce2 6192 |
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