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Mirrors > Home > MPE Home > Th. List > ssun4 | Structured version Visualization version GIF version |
Description: Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
ssun4 | ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 4032 | . 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵) | |
2 | sstr2 3859 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐶 ∪ 𝐵) → 𝐴 ⊆ (𝐶 ∪ 𝐵))) | |
3 | 1, 2 | mpi 20 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3821 ⊆ wss 3823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-v 3411 df-un 3828 df-in 3830 df-ss 3837 |
This theorem is referenced by: ssun 4047 xpsspw 5528 uncmp 21727 volcn 23922 bnj1408 31982 bnj1452 31998 dftrpred3g 32622 pibt2 34168 elrfi 38715 cnvrcl0 39377 |
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