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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 4136 | . 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) | |
2 | sstr2 3962 | . 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 3922 ⊆ wss 3924 |
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 3488 df-un 3929 df-in 3931 df-ss 3940 |
This theorem is referenced by: ssun 4153 ssunsn2 4746 xpsspw 5668 wfrlem15 7955 uncmp 21994 alexsubALTlem3 22640 sxbrsigalem0 31536 bnj1450 32329 altxpsspw 33445 pibt2 34714 superuncl 40017 cnvtrcl0 40076 |
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