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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 3777 | . 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵) | |
2 | sstr2 3610 | . 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 3572 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-in 3581 df-ss 3588 |
This theorem is referenced by: ssun 3792 xpsspw 5233 uncmp 21206 volcn 23374 bnj1408 31104 bnj1452 31120 dftrpred3g 31733 elrfi 37257 cnvrcl0 37932 |
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