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Mirrors > Home > MPE Home > Th. List > ssun3 | Structured version Visualization version GIF version |
Description: Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ssun3 | ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4004 | . 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) | |
2 | sstr2 3835 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐵 ∪ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))) | |
3 | 1, 2 | mpi 20 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3797 ⊆ wss 3799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-v 3417 df-un 3804 df-in 3806 df-ss 3813 |
This theorem is referenced by: ssun 4020 ssunsn2 4577 xpsspw 5468 wfrlem15 7696 uncmp 21578 alexsubALTlem3 22224 sxbrsigalem0 30879 bnj1450 31665 altxpsspw 32624 superuncl 38715 cnvtrcl0 38775 |
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