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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1138 | 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 |
---|---|
bnj1138.1 | ⊢ 𝐴 = (𝐵 ∪ 𝐶) |
Ref | Expression |
---|---|
bnj1138 | ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1138.1 | . . 3 ⊢ 𝐴 = (𝐵 ∪ 𝐶) | |
2 | 1 | eleq2i 2901 | . 2 ⊢ (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ (𝐵 ∪ 𝐶)) |
3 | elun 4122 | . 2 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) | |
4 | 2, 3 | bitri 276 | 1 ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-un 3938 |
This theorem is referenced by: bnj1424 32009 bnj1408 32205 bnj1417 32210 |
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