![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ssun4 | 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 3165 | . 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵) | |
2 | sstr2 3033 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐶 ∪ 𝐵) → 𝐴 ⊆ (𝐶 ∪ 𝐵))) | |
3 | 1, 2 | mpi 15 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∪ cun 2998 ⊆ wss 3000 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 |
This theorem is referenced by: ssun 3180 xpsspw 4563 |
Copyright terms: Public domain | W3C validator |