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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 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | difexi 5330 | . . 3 ⊢ (𝑥 ∖ 𝑦) ∈ V | 
| 4 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 5 | 4 | difexi 5330 | . . 3 ⊢ (𝑦 ∖ 𝑥) ∈ V | 
| 6 | 3, 5 | unex 7764 | . 2 ⊢ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ V | 
| 7 | sseq1 4009 | . 2 ⊢ (𝑧 = ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) → (𝑧 ⊆ 𝐵 ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)) | |
| 8 | sseq1 4009 | . 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
| 9 | sseq1 4009 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵)) | |
| 10 | ssdifss 4140 | . . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∖ 𝑦) ⊆ 𝐵) | |
| 11 | ssdifss 4140 | . . 3 ⊢ (𝑦 ⊆ 𝐵 → (𝑦 ∖ 𝑥) ⊆ 𝐵) | |
| 12 | unss 4190 | . . . 4 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵) | |
| 13 | 12 | biimpi 216 | . . 3 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵) | 
| 14 | 10, 11, 13 | syl2an 596 | . 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵) | 
| 15 | 1, 6, 7, 8, 9, 14 | cllem0 43579 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 | 
| This theorem is referenced by: (None) | 
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