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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-unexg | Structured version Visualization version GIF version | ||
| Description: Existence of binary unions of sets, proved from ax-bj-bun 37038. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-unexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elissetv 2822 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
| 2 | elissetv 2822 | . 2 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵) | |
| 3 | exdistrv 1955 | . . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) | |
| 4 | uneq12 4163 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
| 5 | ax-bj-bun 37038 | . . . . . . . . 9 ⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)) | |
| 6 | 5 | spi 2184 | . . . . . . . 8 ⊢ ∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)) | 
| 7 | 6 | spi 2184 | . . . . . . 7 ⊢ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)) | 
| 8 | bj-axbun 37037 | . . . . . . 7 ⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))) | |
| 9 | 7, 8 | mpbir 231 | . . . . . 6 ⊢ (𝑥 ∪ 𝑦) ∈ V | 
| 10 | 4, 9 | eqeltrrdi 2850 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∪ 𝐵) ∈ V) | 
| 11 | 10 | exlimiv 1930 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∪ 𝐵) ∈ V) | 
| 12 | 11 | exlimiv 1930 | . . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∪ 𝐵) ∈ V) | 
| 13 | 3, 12 | sylbir 235 | . 2 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (𝐴 ∪ 𝐵) ∈ V) | 
| 14 | 1, 2, 13 | syl2an 596 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∀wal 1538 = wceq 1540 ∃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-12 2177 ax-ext 2708 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 | 
| This theorem is referenced by: bj-prexg 37040 bj-prex 37041 bj-adjfrombun 37047 | 
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