| Mathbox for Jonathan Ben-Naim |
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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 4995 | . 2 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵) | |
| 3 | 1, 2 | sylib 218 | 1 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∃wrex 3070 ∪ ciun 4991 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-iun 4993 |
| This theorem is referenced by: bnj1408 35050 bnj1450 35064 bnj1501 35081 |
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