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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 37019). (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 4162 | . 2 ⊢ (𝑡 ∈ (𝑥 ∪ 𝑦) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)) | |
2 | 1 | bj-clex 37013 | 1 ⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∨ wo 847 ∀wal 1534 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 ∪ cun 3960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-un 3967 |
This theorem is referenced by: bj-unexg 37020 |
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