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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-prfromadj | Structured version Visualization version GIF version |
Description: Unordered pair from adjunction. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-prfromadj | ⊢ {𝑥, 𝑦} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4625 | . 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦}) | |
2 | bj-snfromadj 35729 | . . 3 ⊢ {𝑥} ∈ V | |
3 | bj-adjg1 35728 | . . 3 ⊢ ({𝑥} ∈ V → ({𝑥} ∪ {𝑦}) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ({𝑥} ∪ {𝑦}) ∈ V |
5 | 1, 4 | eqeltri 2828 | 1 ⊢ {𝑥, 𝑦} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3473 ∪ cun 3942 {csn 4622 {cpr 4624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2702 ax-nul 5299 ax-bj-adj 35727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-dif 3947 df-un 3949 df-nul 4319 df-sn 4623 df-pr 4625 |
This theorem is referenced by: (None) |
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