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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1424 | 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 |
|---|---|
| bnj1424.1 | ⊢ 𝐴 = (𝐵 ∪ 𝐶) |
| Ref | Expression |
|---|---|
| bnj1424 | ⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1424.1 | . . 3 ⊢ 𝐴 = (𝐵 ∪ 𝐶) | |
| 2 | 1 | bnj1138 34802 | . 2 ⊢ (𝐷 ∈ 𝐴 ↔ (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶)) |
| 3 | 2 | biimpi 216 | 1 ⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∪ 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: bnj1423 35065 bnj1452 35066 |
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