Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > clel2 | Structured version Visualization version GIF version |
Description: Alternate definition of membership when the member 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 3587 | . 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 205 ∀wal 1537 = wceq 1539 ∈ wcel 2106 Vcvv 3429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 |
This theorem is referenced by: mptelee 27273 |
Copyright terms: Public domain | W3C validator |