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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssuncl | Structured version Visualization version GIF version |
Description: The class of all subsets of a class is closed under binary union. (Contributed by RP, 3-Jan-2020.) |
Ref | Expression |
---|---|
ssficl.a | ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵} |
Ref | Expression |
---|---|
ssuncl | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssficl.a | . 2 ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵} | |
2 | vex 3470 | . . 3 ⊢ 𝑥 ∈ V | |
3 | vex 3470 | . . 3 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | unex 7727 | . 2 ⊢ (𝑥 ∪ 𝑦) ∈ V |
5 | sseq1 4000 | . 2 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵)) | |
6 | sseq1 4000 | . 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
7 | sseq1 4000 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵)) | |
8 | unss 4177 | . . 3 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵) | |
9 | 8 | biimpi 215 | . 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∪ 𝑦) ⊆ 𝐵) |
10 | 1, 4, 5, 6, 7, 9 | cllem0 42867 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2701 ∀wral 3053 Vcvv 3466 ∪ cun 3939 ⊆ wss 3941 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
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 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-sn 4622 df-pr 4624 df-uni 4901 |
This theorem is referenced by: (None) |
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