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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 3815 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
2 | unss2 3817 | . 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) | |
3 | 1, 2 | sylan9ss 3649 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∪ cun 3605 ⊆ wss 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-v 3233 df-un 3612 df-in 3614 df-ss 3621 |
This theorem is referenced by: pwssun 5049 fun 6104 undom 8089 finsschain 8314 trclun 13799 relexpfld 13833 mvdco 17911 dprd2da 18487 dmdprdsplit2lem 18490 lspun 19035 spanuni 28531 sshhococi 28533 mthmpps 31605 mblfinlem3 33578 dochdmj1 36996 mptrcllem 38237 clcnvlem 38247 dfrcl2 38283 relexpss1d 38314 corclrcl 38316 relexp0a 38325 corcltrcl 38348 frege131d 38373 |
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