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Mirrors > Home > ILE Home > Th. List > unissd | GIF version |
Description: Subclass relationship for subclass union. Deduction form of uniss 3765. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
unissd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
unissd | ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unissd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | uniss 3765 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3076 ∪ cuni 3744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 df-uni 3745 |
This theorem is referenced by: iotanul 5111 tfrlemibfn 6233 tfrlemiubacc 6235 tfr1onlemssrecs 6244 tfr1onlembfn 6249 tfr1onlemubacc 6251 tfrcllemssrecs 6257 tfrcllembfn 6262 tfrcllemubacc 6264 fiuni 6874 eltg3i 12264 unitg 12270 tgss 12271 ntrss 12327 |
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