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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssdifcl | Structured version Visualization version GIF version |
Description: The class of all subsets of a class is closed under class difference. (Contributed by RP, 3-Jan-2020.) |
Ref | Expression |
---|---|
ssficl.a | ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵} |
Ref | Expression |
---|---|
ssdifcl | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∖ 𝑦) ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssficl.a | . 2 ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵} | |
2 | vex 3498 | . . 3 ⊢ 𝑥 ∈ V | |
3 | 2 | difexi 5225 | . 2 ⊢ (𝑥 ∖ 𝑦) ∈ V |
4 | sseq1 3992 | . 2 ⊢ (𝑧 = (𝑥 ∖ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∖ 𝑦) ⊆ 𝐵)) | |
5 | sseq1 3992 | . 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
6 | sseq1 3992 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵)) | |
7 | ssdifss 4112 | . . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∖ 𝑦) ⊆ 𝐵) | |
8 | 7 | adantr 483 | . 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∖ 𝑦) ⊆ 𝐵) |
9 | 1, 3, 4, 5, 6, 8 | cllem0 39918 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∖ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 Vcvv 3495 ∖ cdif 3933 ⊆ wss 3936 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 |
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-ral 3143 df-rab 3147 df-v 3497 df-dif 3939 df-in 3943 df-ss 3952 |
This theorem is referenced by: (None) |
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