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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 4113 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
2 | uncom 4087 | . 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶) | |
3 | uncom 4087 | . 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶) | |
4 | 1, 2, 3 | 3sstr4g 3966 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3885 ⊆ wss 3887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-un 3892 df-in 3894 df-ss 3904 |
This theorem is referenced by: unss12 4116 ord3ex 5310 xpider 8577 fin1a2lem13 10168 canthp1lem2 10409 seqexw 13737 uniioombllem3 24749 volcn 24770 dvres2lem 25074 bnj1413 33015 bnj1408 33016 |
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