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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 3495 | . . 3 ⊢ 𝑥 ∈ V | |
3 | vex 3495 | . . 3 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | unex 7458 | . 2 ⊢ (𝑥 ∪ 𝑦) ∈ V |
5 | sseq1 3989 | . 2 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵)) | |
6 | sseq1 3989 | . 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
7 | sseq1 3989 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵)) | |
8 | unss 4157 | . . 3 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵) | |
9 | 8 | biimpi 217 | . 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∪ 𝑦) ⊆ 𝐵) |
10 | 1, 4, 5, 6, 7, 9 | cllem0 39803 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ∀wral 3135 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-pr 4560 df-uni 4831 |
This theorem is referenced by: (None) |
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