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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1405 | 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 |
---|---|
bnj1405.1 | ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵) |
Ref | Expression |
---|---|
bnj1405 | ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1405.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵) | |
2 | eliun 4460 | . 2 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵) | |
3 | 1, 2 | sylib 207 | 1 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∃wrex 2897 ∪ ciun 4455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-iun 4457 |
This theorem is referenced by: bnj1408 30358 bnj1450 30372 bnj1501 30389 |
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