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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 3340 | . 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 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 |
This theorem is referenced by: unipr 4449 brcup 32046 brcap 32047 |
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