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Mirrors > Home > MPE Home > Th. List > Mathboxes > sssymdifcl | Structured version Visualization version GIF version |
Description: The class of all subsets of a class is closed under symmetric difference. (Contributed by RP, 3-Jan-2020.) |
Ref | Expression |
---|---|
ssficl.a | ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵} |
Ref | Expression |
---|---|
sssymdifcl | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssficl.a | . 2 ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵} | |
2 | vex 3472 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | difexi 5321 | . . 3 ⊢ (𝑥 ∖ 𝑦) ∈ V |
4 | vex 3472 | . . . 4 ⊢ 𝑦 ∈ V | |
5 | 4 | difexi 5321 | . . 3 ⊢ (𝑦 ∖ 𝑥) ∈ V |
6 | 3, 5 | unex 7730 | . 2 ⊢ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ V |
7 | sseq1 4002 | . 2 ⊢ (𝑧 = ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) → (𝑧 ⊆ 𝐵 ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)) | |
8 | sseq1 4002 | . 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
9 | sseq1 4002 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵)) | |
10 | ssdifss 4130 | . . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∖ 𝑦) ⊆ 𝐵) | |
11 | ssdifss 4130 | . . 3 ⊢ (𝑦 ⊆ 𝐵 → (𝑦 ∖ 𝑥) ⊆ 𝐵) | |
12 | unss 4179 | . . . 4 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵) | |
13 | 12 | biimpi 215 | . . 3 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵) |
14 | 10, 11, 13 | syl2an 595 | . 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵) |
15 | 1, 6, 7, 8, 9, 14 | cllem0 42893 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 ∀wral 3055 Vcvv 3468 ∖ cdif 3940 ∪ cun 3941 ⊆ wss 3943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-sn 4624 df-pr 4626 df-uni 4903 |
This theorem is referenced by: (None) |
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