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| Mirrors > Home > MPE Home > Th. List > unss2 | Structured version Visualization version GIF version | ||
| Description: Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.) |
| Ref | Expression |
|---|---|
| unss2 | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unss1 4185 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
| 2 | uncom 4158 | . 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶) | |
| 3 | uncom 4158 | . 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶) | |
| 4 | 1, 2, 3 | 3sstr4g 4037 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∪ 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 |
| This theorem is referenced by: unss12 4188 ord3ex 5387 xpider 8828 fin1a2lem13 10452 canthp1lem2 10693 seqexw 14058 uniioombllem3 25620 volcn 25641 dvres2lem 25945 mulsproplem13 28154 mulsproplem14 28155 bnj1413 35049 bnj1408 35050 |
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