Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4155 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
2 | uncom 4129 | . 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶) | |
3 | uncom 4129 | . 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶) | |
4 | 1, 2, 3 | 3sstr4g 4012 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3934 ⊆ wss 3936 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-un 3941 df-in 3943 df-ss 3952 |
This theorem is referenced by: unss12 4158 ord3ex 5280 xpider 8362 fin1a2lem13 9828 canthp1lem2 10069 seqexw 13379 uniioombllem3 24180 volcn 24201 dvres2lem 24502 bnj1413 32302 bnj1408 32303 |
Copyright terms: Public domain | W3C validator |