Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rabswap | Structured version Visualization version GIF version |
Description: Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.) |
Ref | Expression |
---|---|
rabswap | ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 463 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
2 | 1 | rabbia2 3477 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {crab 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1536 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-rab 3147 |
This theorem is referenced by: incom 4177 |
Copyright terms: Public domain | W3C validator |