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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-axbun | Structured version Visualization version GIF version | ||
| Description: Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 37038). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-axbun | ⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elun 4153 | . 2 ⊢ (𝑡 ∈ (𝑥 ∪ 𝑦) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)) | |
| 2 | 1 | bj-clex 37032 | 1 ⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 848 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 | 
| 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-ext 2708 | 
| 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 | 
| This theorem is referenced by: bj-unexg 37039 | 
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