Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > clel2g | 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.) (Revised by BJ, 12-Feb-2022.) |
Ref | Expression |
---|---|
clel2g | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1914 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | |
2 | eleq1 2903 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | ceqsalg 3532 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
4 | 3 | bicomd 225 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 = wceq 1536 ∈ wcel 2113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1085 df-ex 1780 df-nf 1784 df-cleq 2817 df-clel 2896 |
This theorem is referenced by: clel2 3656 snssg 4720 |
Copyright terms: Public domain | W3C validator |