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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 4925 | . 2 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵) | |
3 | 1, 2 | sylib 220 | 1 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∃wrex 3141 ∪ ciun 4921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-iun 4923 |
This theorem is referenced by: bnj1408 32310 bnj1450 32324 bnj1501 32341 |
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