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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 4195 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
2 | uncom 4168 | . 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶) | |
3 | uncom 4168 | . 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶) | |
4 | 1, 2, 3 | 3sstr4g 4041 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3961 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 |
This theorem is referenced by: unss12 4198 ord3ex 5393 xpider 8827 fin1a2lem13 10450 canthp1lem2 10691 seqexw 14055 uniioombllem3 25634 volcn 25655 dvres2lem 25960 mulsproplem13 28169 mulsproplem14 28170 bnj1413 35028 bnj1408 35029 |
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