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Mathbox for Jonathan Ben-Naim |
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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 34780 | . 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 847 = wceq 1536 ∈ wcel 2105 ∪ 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: bnj1423 35043 bnj1452 35044 |
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