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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 4087 | . 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵) | |
2 | sstr2 3908 | . 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 3864 ⊆ wss 3866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-un 3871 df-in 3873 df-ss 3883 |
This theorem is referenced by: ssun 4103 xpsspw 5679 dftrpred3g 9339 uncmp 22300 volcn 24503 bnj1408 32729 bnj1452 32745 pibt2 35325 elrfi 40219 cnvrcl0 40909 |
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