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Mirrors > Home > ILE Home > Th. List > unssd | GIF version |
Description: A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
unssd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
unssd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
unssd | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unssd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
2 | unssd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | unss 3117 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
4 | 3 | biimpi 113 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
5 | 1, 2, 4 | syl2anc 391 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∪ cun 2915 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 |
This theorem is referenced by: tpssi 3530 un0addcl 8215 un0mulcl 8216 fzosplit 9033 fzouzsplit 9035 bj-omtrans 10081 |
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