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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 453 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
2 | 1 | abbii 2839 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)} |
3 | df-rab 3092 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
4 | df-rab 3092 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)} | |
5 | 2, 3, 4 | 3eqtr4i 2807 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1508 ∈ wcel 2051 {cab 2753 {crab 3087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-9 2060 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1744 df-sb 2017 df-clab 2754 df-cleq 2766 df-rab 3092 |
This theorem is referenced by: (None) |
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