Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dfsymdif4 | Structured version Visualization version GIF version |
Description: Alternate definition of the symmetric difference. (Contributed by NM, 17-Aug-2004.) (Revised by AV, 17-Aug-2022.) |
Ref | Expression |
---|---|
dfsymdif4 | ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsymdif 4223 | . 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | abbi2i 2953 | 1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1533 ∈ wcel 2110 {cab 2799 △ csymdif 4217 |
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-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-un 3940 df-symdif 4218 |
This theorem is referenced by: mbfeqalem1 24236 |
Copyright terms: Public domain | W3C validator |