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Mirrors > Home > ILE Home > Th. List > unissd | GIF version |
Description: Subclass relationship for subclass union. Deduction form of uniss 3680. (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 3680 | . 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 3000 ∪ cuni 3659 |
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-in 3006 df-ss 3013 df-uni 3660 |
This theorem is referenced by: iotanul 5008 tfrlemibfn 6107 tfrlemiubacc 6109 tfr1onlemssrecs 6118 tfr1onlembfn 6123 tfr1onlemubacc 6125 tfrcllemssrecs 6131 tfrcllembfn 6136 tfrcllemubacc 6138 eltg3i 11810 unitg 11816 tgss 11817 ntrss 11873 |
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