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Mirrors > Home > MPE Home > Th. List > clel3 | Structured version Visualization version GIF version |
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
clel3.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
clel3 | ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clel3.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | clel3g 3602 | . 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: unipr 4817 brcup 33513 brcap 33514 |
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