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Mirrors > Home > MPE Home > Th. List > unss12 | Structured version Visualization version GIF version |
Description: Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.) |
Ref | Expression |
---|---|
unss12 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unss1 4155 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
2 | unss2 4157 | . 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) | |
3 | 1, 2 | sylan9ss 3980 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∪ cun 3934 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3941 df-in 3943 df-ss 3952 |
This theorem is referenced by: pwssun 5456 fun 6540 undom 8605 finsschain 8831 trclun 14374 relexpfld 14408 mulgfval 18226 mvdco 18573 dprd2da 19164 dmdprdsplit2lem 19167 lspun 19759 spanuni 29321 sshhococi 29323 mthmpps 32829 pibt2 34701 mblfinlem3 34946 dochdmj1 38541 mptrcllem 39993 clcnvlem 40003 dfrcl2 40039 relexpss1d 40070 corclrcl 40072 relexp0a 40081 corcltrcl 40104 frege131d 40129 |
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