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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1149 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bnj1149.1 | ⊢ (𝜑 → 𝐴 ∈ V) | 
| bnj1149.2 | ⊢ (𝜑 → 𝐵 ∈ V) | 
| Ref | Expression | 
|---|---|
| bnj1149 | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj1149.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
| 2 | bnj1149.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) | |
| 3 | unexg 7764 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-sn 4626 df-pr 4628 df-uni 4907 | 
| This theorem is referenced by: bnj1136 35012 bnj1413 35050 bnj1452 35067 bnj1489 35071 | 
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