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| Mirrors > Home > MPE Home > Th. List > elex22 | Structured version Visualization version GIF version | ||
| Description: If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.) | 
| Ref | Expression | 
|---|---|
| elex22 | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eleq1a 2836 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
| 2 | eleq1a 2836 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐶)) | |
| 3 | 1, 2 | anim12ii 618 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) | 
| 4 | 3 | alrimiv 1927 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) | 
| 5 | elissetv 2822 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
| 6 | 5 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥 𝑥 = 𝐴) | 
| 7 | exim 1834 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) | |
| 8 | 4, 6, 7 | sylc 65 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 | 
| 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-ex 1780 df-cleq 2729 df-clel 2816 | 
| This theorem is referenced by: en3lplem1VD 44863 | 
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