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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 1588 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | df-rex 2423 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) | |
3 | df-clel 2136 | . 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 1332 ∃wex 1469 ∈ wcel 1481 ∃wrex 2418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-clel 2136 df-rex 2423 |
This theorem is referenced by: reueq 2887 reuind 2893 0el 3390 iunid 3876 sucel 4340 reusv3 4389 fvmptt 5520 releldm2 6091 qsid 6502 rerecclap 8514 nndiv 8785 zq 9445 4fvwrd4 9948 bj-bdcel 13206 |
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