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Mirrors > Home > ILE Home > Th. List > elex22 | 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 2150 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | eleq1a 2150 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐶)) | |
3 | 1, 2 | anim12ii 335 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
4 | 3 | alrimiv 1795 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
5 | elisset 2613 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
6 | 5 | adantr 270 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥 𝑥 = 𝐴) |
7 | exim 1530 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) | |
8 | 4, 6, 7 | sylc 61 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 = wceq 1284 ∃wex 1421 ∈ wcel 1433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 |
This theorem is referenced by: (None) |
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