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Mirrors > Home > MPE Home > Th. List > clel4 | 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 |
---|---|
clel4.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
clel4 | ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clel4.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | eleq2 2898 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | ceqsalv 3530 | . 2 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥) ↔ 𝐴 ∈ 𝐵) |
4 | 3 | bicomi 225 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 = wceq 1528 ∈ wcel 2105 Vcvv 3492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-3an 1081 df-ex 1772 df-nf 1776 df-cleq 2811 df-clel 2890 |
This theorem is referenced by: intpr 4900 |
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