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Mirrors > Home > ILE Home > Th. List > risset | GIF version |
Description: Two ways to say "𝐴 belongs to 𝐵." (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
risset | ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1555 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | df-rex 2381 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) | |
3 | df-clel 2096 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
4 | 1, 2, 3 | 3bitr4ri 212 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1299 ∃wex 1436 ∈ wcel 1448 ∃wrex 2376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-4 1455 ax-ial 1482 |
This theorem depends on definitions: df-bi 116 df-clel 2096 df-rex 2381 |
This theorem is referenced by: reueq 2836 reuind 2842 0el 3332 iunid 3815 sucel 4270 reusv3 4319 fvmptt 5444 releldm2 6013 qsid 6424 rerecclap 8351 nndiv 8619 zq 9268 4fvwrd4 9758 bj-bdcel 12616 |
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