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Mirrors > Home > MPE Home > Th. List > clel2 | 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 |
---|---|
clel2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
clel2 | ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clel2.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | clel2g 3489 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1629 = wceq 1631 ∈ wcel 2145 Vcvv 3351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-12 2203 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-v 3353 |
This theorem is referenced by: mptelee 25996 |
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