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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 3282 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
4 | 3 | biimpi 119 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
5 | 1, 2, 4 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∪ cun 3100 ⊆ wss 3102 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 |
This theorem is referenced by: tpssi 3724 casef 7035 un0addcl 9129 un0mulcl 9130 fzosplit 10086 fzouzsplit 10088 exmidunben 12251 strleund 12374 fsumcncntop 13052 bj-charfun 13479 bj-omtrans 13628 |
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