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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-adjfrombun | Structured version Visualization version GIF version | ||
| Description: Adjunction from singleton and binary union. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-adjfrombun | ⊢ (𝑥 ∪ {𝑦}) ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . 2 ⊢ 𝑥 ∈ V | |
| 2 | bj-snexg 37035 | . . 3 ⊢ (𝑦 ∈ V → {𝑦} ∈ V) | |
| 3 | 2 | elv 3485 | . 2 ⊢ {𝑦} ∈ V |
| 4 | bj-unexg 37039 | . 2 ⊢ ((𝑥 ∈ V ∧ {𝑦} ∈ V) → (𝑥 ∪ {𝑦}) ∈ V) | |
| 5 | 1, 3, 4 | mp2an 692 | 1 ⊢ (𝑥 ∪ {𝑦}) ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-bj-sn 37034 ax-bj-bun 37038 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 |
| This theorem is referenced by: (None) |
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