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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 3744 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
2 | uncom 3719 | . 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶) | |
3 | uncom 3719 | . 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶) | |
4 | 1, 2, 3 | 3sstr4g 3609 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3538 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 |
This theorem is referenced by: unss12 3747 ord3ex 4782 xpider 7705 fin1a2lem13 9117 canthp1lem2 9354 uniioombllem3 23159 volcn 23180 dvres2lem 23480 bnj1413 30357 bnj1408 30358 |
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